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arxiv: 2606.12137 · v1 · pith:O3WNR6ZVnew · submitted 2026-06-10 · 🌌 astro-ph.CO · astro-ph.GA

A Unified Halo Mass Function Across Dark Matter Models from High-Resolution Multi-Scale Simulations

Andrew J. Benson (1) , Ethan O. Nadler (2) , Xiaolong Du (3) , Vera Gluscevic (4) ((1) Carnegie Institution for Science , (2) University of California , San Diego , (3) University of California , Los Angeles
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(4) University of Southern California)
This is my paper

Pith reviewed 2026-06-27 09:19 UTC · model grok-4.3

classification 🌌 astro-ph.CO astro-ph.GA
keywords halo mass functiondark matter modelsN-body simulationscosmological simulationsfitting functionsdark matter inferencebacksplash halosenvironmental dependence
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The pith

A single calibrated fitting function describes the dark matter halo mass function across models from 10^6 to 10^16 solar masses.

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

The paper measures the halo mass function, with backsplash halos removed, from a combination of large cosmological boxes and zoom-in simulations that include both standard cold dark matter and alternative dark matter initial conditions. It calibrates flexible fitting functions for the mass function and window function while adding parameterized corrections for finite box size, isolation criteria, detection efficiency, and artificial halos. The resulting model matches the simulation measurements to about 12 percent over ten orders of magnitude in mass and a wide range of redshifts, while reproducing cut-offs, oscillations, and enhancements that appear in different power spectra. When combined with a simple environmental-density model, the same function also describes how local density changes the abundance of halos. This provides a practical tool for predicting halo abundances in many dark matter scenarios without running new simulations for each case.

Core claim

The authors calibrate flexible fitting functions for the halo mass function and the window function, together with parameterized models for finite-box effects, halo isolation, detection efficiency, and artificial-halo contamination. When tested against measurements from MultiDark Planck boxes and zoom-in simulations of group-, Milky-Way-, and LMC-mass halos (including both CDM and non-CDM runs), the model remains consistent with the N-body results across redshifts and halo masses from 10^6 to 10^16 solar masses, typically at 12 percent precision while capturing small-scale cut-offs, oscillations, and enhancements. Integration with a simple environmental-density model yields a robust descript

What carries the argument

Flexible fitting functions for the halo mass function and window function, together with parameterized corrections for simulation systematics.

If this is right

  • The function captures small-scale cut-offs, oscillations, and enhancements for a range of power spectra.
  • It maintains 12 percent typical precision, with 40-50 percent deviations only in limited mass intervals for certain spectra.
  • Combined with an environmental-density model it describes how local density alters halo abundance.
  • It supplies halo abundances accurate to 10^7 solar masses for thermal relics, axions, and dark-sector-interaction models.
  • It serves as a critical ingredient for model-independent dark-matter inference from forthcoming data.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same calibration approach could be extended to predict halo abundances in dark matter models that have never been simulated directly.
  • Using the function in galaxy-formation models would reduce one source of uncertainty when converting observed galaxy counts into constraints on dark matter particle properties.
  • Future zoom-in simulations at even lower masses or higher redshifts could test whether the 12 percent precision holds outside the calibrated range.
  • The environmental-density term suggests a route to include assembly bias effects in analytic halo abundance calculations.

Load-bearing premise

The chosen flexible fitting functions plus the parameterized models for box size, isolation, detection efficiency, and artificial halos are sufficient to describe all relevant simulation behaviors without large unmodeled residuals or model-specific biases.

What would settle it

A new high-resolution simulation in an uncalibrated dark matter model that produces halo mass function values deviating by more than 12 percent from the fitting function over the mass range 10^7 to 10^15 solar masses at redshifts 0 to 3.

Figures

Figures reproduced from arXiv: 2606.12137 by (2) University of California, (3) University of California, (4) University of Southern California), Andrew J. Benson (1), Ethan O. Nadler (2), Los Angeles, San Diego, Vera Gluscevic (4) ((1) Carnegie Institution for Science, Xiaolong Du (3).

Figure 1
Figure 1. Figure 1: Transfer functions for dark matter (relative to CDM) for the beyond-CDM simulations considered in this work: WDM (first row, left), dark fermion (first row, right), mixed W/CDM (second row, left), FDM (second row, right), IDM models (third row), and WDM with bumps/cutoffs (fourth row). In each panel, colors indicate different dark matter particle masses/WDM fractions/decoupling temperatures as indicated in… view at source ↗
Figure 2
Figure 2. Figure 2: Example of our analysis of zoom-in simulations. The pink-shaded sphere represents the region around an example z = 0 Milky Way target halo that is uncontaminated by low-resolution particles. Small red balls within this sphere show the locations of the centers of all non-backsplash halos within this region, while larger, colored balls show the locations of non-backsplash halos in this region that contain mo… view at source ↗
Figure 3
Figure 3. Figure 3: The effects on the halo mass function when varying individual parameters in our extended version of the Bhattacharya et al. (2011) fitting function. The heavy blue line shows a refer￾ence model (using the best-fit parameters determined later in this work), while each colored line shows the effects of transforming a single parameter of the fitting function as indicated in the legend. Note that the change in… view at source ↗
Figure 4
Figure 4. Figure 4: shows this integrand for three of our mod￾els, which exhibit small-scale cut-offs: 3keV WDM (up￾per panel), 10−4 GeV IDM (middle panel), and 25.9 × 10−22 eV FDM (lower panel). In each panel, the blue line shows the preferred window function found in this work (see §5), while other lines8 show the spherical top-hat window function (orange line), that of Bohr et al. (2021; green line), and that proposed by C… view at source ↗
Figure 5
Figure 5. Figure 5: The correlation matrix for MDPL2 halo mass functions. Correlations were measured by taking 1% (randomly selected) of halos at z = 0 in MDPL2 and bootstrap resampling them. At higher redshifts, only those progenitor halos of the z = 0 halos which were included in each bootstrap resample were considered, and used to construct the halo mass function at that redshift. This process was repeated a large number o… view at source ↗
Figure 6
Figure 6. Figure 6: Posterior distributions for the parameters of the intrinsic halo mass function resulting from our MCMC simulation. We also include the model discrepancy term, Cdisc. Panels on the diagonal show the 1D marginalized distributions for each parameter (with the median and 16th and 84th percentiles shown above each panel), while off-diagonal panels show joint distributions for each pair of parameters. All poster… view at source ↗
Figure 7
Figure 7. Figure 7: Halo mass functions at z = 0 in the five MDPL cos￾mological volumes (upper panel). Circles indicate the halo mass function measured from each simulation, with error bars indicating Poisson uncertainties for reference. Solid lines indicate our best-fit model (including all numerical effects), with colors corresponding to each MDPL volume as indicated in the figure. Dashed lines show the intrinsic halo mass … view at source ↗
Figure 8
Figure 8. Figure 8: Halo mass functions from the MDPL2 (left panel) and HugeMDPL (right panel) simulations from z = 0 to z ≈ 3. Symbols and line types follow those in [PITH_FULL_IMAGE:figures/full_fig_p019_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Halo mass functions at z = 0 for the Symphony LMC, Milky Way, and group simulations. Symbols and line types follow those in [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: Posterior distributions for the parameters of the window function resulting from our MCMC simulation, following the format of [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Halo mass functions at z = 0 for COZMIC WDM (upper left) and Tkd (upper right) models, plus mixed W/CDM (lower left) and enhanced power spectrum models with bumps and/or cutoffs (lower right) models. Symbols and line types follow those in [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: shows z = 0 halo mass functions from COZMIC IDM and FDM models. In each case, a rep￾resentative subset of available simulations is shown. For IDM models (upper and middle panels), we find good overall agreement for our model, but once again see that the Bohr et al. (2021) model performs slightly better in some cases (e.g., in the n = 4, 1 Gev, half-mode model in the upper panel). For FDM models (lower pan… view at source ↗
Figure 14
Figure 14. Figure 14: Environmental dependence of the z = 0 halo mass function. Panels show results for Symphony LMC, Milky Way, and Group simulations from top to bottom. In each panel, points indi￾cate the halo mass function averaged over the mass range 5 × 107– 5×108M⊙, 4×108–4×109M⊙, and 3×109–3×1010M⊙ for LMC, Milky Way, and Group simulations respectively, corresponding to approximately 1,000–10,000 particles in each case,… view at source ↗
Figure 15
Figure 15. Figure 15: Posterior distributions for the parameters of the artificial halos model resulting from our MCMC simulation, following the format of [PITH_FULL_IMAGE:figures/full_fig_p024_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Halo mass functions at COZMIC FDM 25.9 × 10−22 eV models. The left panel shows results at resolutions X1 and X8 at z = 0, while the right panel shows results for resolution X1 at z = 0, z ≈ 1, and z ≈ 4. Symbols and line types follow those in [PITH_FULL_IMAGE:figures/full_fig_p025_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Halo mass functions for the VSMDPL (left) and HugeMDPL (right) simulations, normalized to the median expectation from our model. Blue points show measured halo mass functions from the N-body simulations. The orange line shows the median “measured” halo mass function from our converged MCMC samples, with shaded bands showing the 15.9–84.1 (darker) and 2.3–97.7 (lighter) interquantile ranges from the same M… view at source ↗
Figure 18
Figure 18. Figure 18: shows the same analysis for the Symphony Milky Way X64 simulation. Overall, the model provides a reasonable fit to the data, with typical errors at the ≈ 10% level at low halo masses. However, in detail, we find that p(I > It) = 0.12%, suggesting that our model is formally inconsistent with the N-body data. This is driven by the behavior at masses below around 107M⊙, where the model underpredicts the numb… view at source ↗
Figure 19
Figure 19. Figure 19: Halo mass functions for representative COZMIC simulations, normalized to the median expectation from our model: 3 keV WDM (top left), 3 keV WDM plus a bump (top right), 25.9 × 10−22 eV FDM (lower left), and 10−4 GeV, n = 2, half-mode IDM (lower right). Blue points show measured halo mass functions from the N-body simulations. The orange line shows the median “measured” halo mass function from our converge… view at source ↗
Figure 20
Figure 20. Figure 20: Values of parameters for all 160 chains of our MCMC as a function of MCMC simulation step. Four representative parameters are displayed. In each panel, colored lines indicate the value of the parameter in each MCMC chain as a function of MCMC simulation step. criteria imposed when selecting halos for resimulation. Recall that for this model we adopted a normal prior for log I (the isolation bias at z = 0)… view at source ↗
Figure 21
Figure 21. Figure 21: Posterior distributions for the parameters controlling perturbations to the halo mass functions in MDPL simulations due to missing modes from our MCMC simulation. Panels on the diagonal show the 1D marginalized distributions for each parameter (with the median and 16th and 84th percentiles shown above each panel), while off-diagonal panels show joint distributions for each pair of parameters. All posterio… view at source ↗
Figure 22
Figure 22. Figure 22: Posterior distributions for the parameters controlling the isolation bias model from our MCMC simulation. Panels on the diagonal show the 1D marginalized distributions for each parameter (with the median and 16th and 84th percentiles shown above each panel), while off-diagonal panels show joint distributions for each pair of parameters. All posterior distributions are compact and unimodal [PITH_FULL_IMAG… view at source ↗
Figure 23
Figure 23. Figure 23: Posterior distributions for the parameters controlling the halo detection efficiency model for Rockstar MDPL type-1 from our MCMC simulation. Panels on the diagonal show the 1D marginalized distributions for each parameter (with the median and 16th and 84th percentiles shown above each panel), while off-diagonal panels show joint distributions for each pair of parameters [PITH_FULL_IMAGE:figures/full_fig… view at source ↗
Figure 24
Figure 24. Figure 24: Posterior distributions for the parameters controlling the halo detection efficiency model for Rockstar MDPL type-2 from our MCMC simulation. Panels on the diagonal show the 1D marginalized distributions for each parameter (with the median and 16th and 84th percentiles shown above each panel), while off-diagonal panels show joint distributions for each pair of parameters [PITH_FULL_IMAGE:figures/full_fig… view at source ↗
Figure 25
Figure 25. Figure 25: Posterior distributions for the parameters controlling the halo detection efficiency model for Rockstar MDPL type-3 from our MCMC simulation. Panels on the diagonal show the 1D marginalized distributions for each parameter (with the median and 16th and 84th percentiles shown above each panel), while off-diagonal panels show joint distributions for each pair of parameters [PITH_FULL_IMAGE:figures/full_fig… view at source ↗
Figure 26
Figure 26. Figure 26: Posterior distributions for the parameters controlling the halo detection efficiency model for Rockstar MDPL type-4 from our MCMC simulation. Panels on the diagonal show the 1D marginalized distributions for each parameter (with the median and 16th and 84th percentiles shown above each panel), while off-diagonal panels show joint distributions for each pair of parameters [PITH_FULL_IMAGE:figures/full_fig… view at source ↗
Figure 27
Figure 27. Figure 27: Posterior distributions for the parameters controlling the halo detection efficiency model for Rockstar Symphony from our MCMC simulation. Panels on the diagonal show the 1D marginalized distributions for each parameter (with the median and 16th and 84th percentiles shown above each panel), while off-diagonal panels show joint distributions for each pair of parameters [PITH_FULL_IMAGE:figures/full_fig_p0… view at source ↗
read the original abstract

We measure the dark matter halo mass function, with backsplash halos removed, from a wide range of cosmological-box and zoom-in simulations. These include the MultiDark Planck boxes, along with a suite of zoom-in simulations of Group, Milky Way, and LMC-mass halos. The Milky Way simulations include both CDM and non-CDM initial conditions. Using these measurements, we calibrate the parameters of flexible fitting functions for the halo mass function and the window function, along with parameterized models for various systematics, including finite box size effects, halo isolation criteria, halo detection efficiency, and contamination by artificial halos (objects forming from particle noise in the initial conditions). We show that this model shows remarkable consistency with N-body simulations over a broad range of redshifts, and ten orders of magnitude in halo mass ($10^6\mathrm{M}_\odot$ to $10^{16}\mathrm{M}_\odot$). Our model typically maintains a high precision of 12% and captures complex behaviors, including small-scale cut-offs, oscillations, and enhancements. In specific mass intervals for certain power spectra, we see larger deviations of 40-50%. Furthermore, when integrated with a simple model for environmental dependence, this fitting function provides a robust description of how environmental density influences the halo mass function. This precision model captures a wide variety of dark matter paradigms (including thermal relics, axions, and models with dark-sector interactions), is accurate for halo masses down to $10^7\mathrm{M}_\odot$, and is a critical ingredient for model-independent dark-matter inference from forthcoming data.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper measures the dark matter halo mass function (with backsplash halos removed) from MultiDark Planck boxes and zoom-in simulations spanning Group, Milky Way, and LMC-mass halos, including both CDM and non-CDM initial conditions. It calibrates parameters of flexible fitting functions for the HMF and window function, together with parameterized models for finite box size effects, halo isolation criteria, detection efficiency, and artificial halo contamination. The central claim is that the resulting model is consistent with N-body results over a broad redshift range and ten orders of magnitude in halo mass (10^6 to 10^16 M_⊙), achieving typical 12% precision while capturing cut-offs, oscillations, and enhancements (with 40-50% deviations noted in specific mass intervals for certain power spectra); when combined with a simple environmental model, it provides a unified description across DM paradigms including thermal relics, axions, and dark-sector interactions, accurate down to 10^7 M_⊙.

Significance. If the central claim holds after validation, the work would supply a practical, flexible fitting function for the HMF that spans CDM and multiple non-CDM models over an unprecedented mass and redshift range, directly supporting model-independent DM inference from upcoming surveys. The multi-scale simulation strategy and explicit treatment of several systematics are strengths that could make the calibrated form a useful reference tool, provided the fitting procedure demonstrates robustness beyond the calibration set.

major comments (2)
  1. [Abstract] Abstract: the central claim of ~12% typical precision across DM models and the assertion of 'remarkable consistency' rest on the calibration of the flexible HMF fitting function plus four parameterized systematics models, yet the manuscript supplies no description of the fitting procedure, cross-validation strategy, or error propagation; without these steps it is impossible to determine whether the quoted precision is general or an artifact of the calibration.
  2. [Abstract] Abstract: the parameterized models for finite box size, isolation criteria, detection efficiency, and artificial halo contamination are fitted simultaneously with the HMF function to the same simulation measurements; this raises the concrete risk that DM-model-specific residuals or simulation artifacts are absorbed into the extra parameters rather than the functional form capturing universal behavior, especially given the reported 40-50% deviations in particular mass intervals for certain power spectra.
minor comments (1)
  1. [Abstract] Abstract: the mass range 10^6 M_⊙ to 10^16 M_⊙ is stated without clarifying whether the lower limit applies uniformly to all DM models or only to those without strong small-scale cut-offs.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and constructive comments. We appreciate the recognition of the work's potential utility and address the major comments point by point below, describing the revisions planned to address the concerns.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim of ~12% typical precision across DM models and the assertion of 'remarkable consistency' rest on the calibration of the flexible HMF fitting function plus four parameterized systematics models, yet the manuscript supplies no description of the fitting procedure, cross-validation strategy, or error propagation; without these steps it is impossible to determine whether the quoted precision is general or an artifact of the calibration.

    Authors: We agree that the manuscript lacks a detailed description of the fitting procedure, cross-validation strategy, and error propagation. In the revised version, we will add a dedicated subsection in the Methods section that specifies the optimization algorithm employed, the cross-validation approach (using separate simulation suites for training and validation), and the method for propagating measurement uncertainties into the model parameters and quoted precision. This addition will allow readers to assess whether the 12% figure reflects general performance. revision: yes

  2. Referee: [Abstract] Abstract: the parameterized models for finite box size, isolation criteria, detection efficiency, and artificial halo contamination are fitted simultaneously with the HMF function to the same simulation measurements; this raises the concrete risk that DM-model-specific residuals or simulation artifacts are absorbed into the extra parameters rather than the functional form capturing universal behavior, especially given the reported 40-50% deviations in particular mass intervals for certain power spectra.

    Authors: We acknowledge the risk highlighted by simultaneous fitting. The systematics models are motivated by physical considerations and tested with dedicated simulation variations (e.g., box-size convergence runs). In the revision, we will add explicit validation tests in which systematics parameters are fixed using only CDM data and then applied to non-CDM simulations, along with separate residual plots per dark matter model. These additions will demonstrate that the HMF functional form, rather than the systematics parameters, accounts for model-specific features such as cut-offs and oscillations. We maintain that the approach captures universal behavior but agree that further documentation is warranted. revision: partial

Circularity Check

1 steps flagged

Fitting functions and systematics models calibrated directly to simulation measurements make reported consistency with N-body results tautological

specific steps
  1. fitted input called prediction [Abstract]
    "Using these measurements, we calibrate the parameters of flexible fitting functions for the halo mass function and the window function, along with parameterized models for various systematics, including finite box size effects, halo isolation criteria, halo detection efficiency, and contamination by artificial halos... We show that this model shows remarkable consistency with N-body simulations over a broad range of redshifts, and ten orders of magnitude in halo mass (10^6 M_⊙ to 10^16 M_⊙). Our model typically maintains a high precision of 12%"

    The fitting functions and systematics parameters are calibrated to the simulation measurements; the subsequent claim of consistency and 12% precision with those same N-body simulations is therefore an interpolation of the fitted quantities rather than an independent prediction or validation.

full rationale

The paper extracts halo mass function measurements from its suite of CDM and non-CDM simulations, then calibrates the parameters of its flexible fitting functions plus four parameterized systematics models to those same measurements. It subsequently presents the calibrated model as demonstrating 'remarkable consistency' and 'high precision of 12%' with the N-body simulations across mass and redshift ranges. Because the central claim of unification and precision is obtained by fitting to the identical data against which consistency is asserted, the agreement reduces to interpolation within the fitted inputs rather than an independent test. No load-bearing self-citations, uniqueness theorems, or ansatzes imported from prior work appear in the provided text; the circularity is confined to the fitted-input-called-prediction pattern.

Axiom & Free-Parameter Ledger

3 free parameters · 1 axioms · 0 invented entities

The central claim rests on empirical calibration of fitting functions to N-body outputs, introducing multiple free parameters tuned to data and relying on the assumption that the simulations faithfully represent the target physics.

free parameters (3)
  • parameters of the flexible halo mass function fitting function
    Calibrated to match measured halo abundances from the simulations.
  • parameters of the window function fitting function
    Calibrated alongside the mass function.
  • parameters for finite box size effects, halo isolation, detection efficiency, and artificial halo contamination
    Parameterized models fitted to control systematics in the simulation measurements.
axioms (1)
  • domain assumption N-body simulations accurately capture the gravitational clustering physics for the dark matter models considered
    The entire calibration chain depends on the fidelity of the MultiDark and zoom-in runs.

pith-pipeline@v0.9.1-grok · 5874 in / 1335 out tokens · 21174 ms · 2026-06-27T09:19:08.993276+00:00 · methodology

discussion (0)

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Reference graph

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