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The halo mass function can be predicted to about 1% at Δ=200, and to about 3% on unseen cosmologies, once non-universality is modelled with the integrated growth history and the local spectral slope.

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2026-08-03 21:03 UTC pith:LSZM3VGK

load-bearing objection A genuinely useful Δ=200 HMF fitting function, but the advertised virial interpolation is not reproducible as written because Eq. (15) goes undefined below Δ≈135. the 2 major comments →

arxiv 2511.16730 v1 pith:LSZM3VGK submitted 2025-11-20 astro-ph.CO

Evolution mapping III: A new recipe for the halo mass function

Andrea Fiorilli , Andr\'es N. Ruiz , Ariel G. Sanchez , Matteo Esposito This is my paper
classification astro-ph.CO PACS 98.80.-k
keywords halo mass functionhalo multiplicity functionevolution mappingnon-universalitystructure formation historyN-body simulationsoverdensity thresholdcosmological parameter estimation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper claims that the dark matter halo mass function—the count of collapsed haloes as a function of mass—can be predicted to about one percent accuracy once two physical inputs are accounted for: the recent integrated growth history of structure and the local slope of the linear power spectrum. Building on the Evolution Mapping framework, the authors introduce an integrated growth parameter x̃ defined as a Gaussian-smoothed integral of Ωm(z)/f²(z) over the past, plus a memory scale η that is fitted rather than fixed. They calibrate a nine-parameter universal functional form for the multiplicity function f(ν) against N-body simulations for ten overdensity thresholds between 150 and 1600, and supply interpolation formulae so the parameters can be computed for any threshold. The central result is per-cent-level accuracy at the reference Δ=200 across masses, redshifts and growth histories, with accuracy within about 3 percent on uncalibrated cosmologies with different power-spectrum shapes, and about 5 percent for other mass definitions including virial masses via interpolation. If correct, this gives cluster abundance analyses a high-precision, physically motivated tool.

Core claim

The paper's central claim is that the previously known non-universality of the halo multiplicity function—its residual dependence on cosmology and time beyond the peak height ν—can be described almost entirely by two variables: x̃, the integrated growth-history parameter, and n_eff, the effective spectral slope of the linear power spectrum at the characteristic peak height ν=1. The proposed model writes f(ν)=A0ν(Aν^a+Bν^b)exp(-Cν²), where A, B, a and b are linear functions of n_eff and x̃, and the memory scale η of the Gaussian kernel in x̃ is a free parameter. Calibrated to N-body simulations spanning very different expansion histories and power-spectrum shapes, the model reproduces the sim

What carries the argument

The central object is the integrated growth-history parameter x̃, defined as x̃(τ|η)=∫_{-∞}^{τ} dτ' x(τ') G(τ'-τ|η), with τ=ln σ12, x=Ωm(z)/f²(z), and G a one-sided Gaussian kernel of width η. It condenses the recent evolution of the matter density and linear growth rate into a single number, capturing differences between cosmologies that share the same linear power spectrum at the same clustering amplitude. The model feeds x̃, together with the effective spectral index n_eff at ν=1, into a universal functional form f(ν)=A0ν(Aν^a+Bν^b)exp(-Cν²), with parameters A=1+An n_eff, B=B0+Bx x̃+Bn n_eff, a=an n_eff, b=bn n_eff. The nine calibrated parameters include the memory scale η, which the data

Load-bearing premise

The load-bearing premise is that a single fitted memory scale η per overdensity threshold—and interpolated all parameters between thresholds—can fully encode how the halo mass function depends on the entire growth history; the systematic residual trends and reversed model ordering in the virial-mass test indicate this premise is only approximately met.

What would settle it

A concrete test: measure the halo mass function at Δ=200 in two cosmologies that have identical linear power spectra at the same σ12 but whose x(τ) histories cross (for example, one with early versus late dark-energy onset), and check whether the model with one η can match both simultaneously to better than a few percent. Alternatively, fit the model parameters directly to virial-mass haloes and compare the best-fit η and residuals with the interpolated predictions; if direct fitting yields a significantly different η or clearly smaller residuals, the interpolation scheme fails.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • Cluster abundance analyses can adopt the Δ=200 recipe with roughly 1% accuracy in the mass function, directly reducing a dominant systematic in cosmological parameter constraints.
  • The model's accuracy on uncalibrated cosmologies with different power-spectrum shapes (within about 3%) means it can be applied without retuning to a broad family of ΛCDM-like models.
  • The continuous interpolation in Δ extends the recipe to any mass definition in the range 150–1600, and even to virial masses outside the calibrated range, with accuracy within about 5%.
  • Compared to existing recipes, the model captures non-universal features—the spread of the mass function across different growth histories at fixed linear power spectrum—that other formulations miss or reproduce only by introducing spurious redshift dependence.

Where Pith is reading between the lines

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

  • The mild trends and model-ordering reversal seen when interpolating to virial masses suggest that one memory scale per threshold is an approximation; a natural extension is to let the Gaussian width η vary with peak height, or to include a second moment of the x(τ) history, which could restore accuracy for arbitrary mass definitions.
  • Because the model already reduces the non-universality of the HMF to the same x̃ that explains power-spectrum non-universality, a unified description of halo abundance, bias and concentration may be within reach—each observable sharing the same x̃ argument.
  • The recipe's algebraic form makes it a natural basis for an emulator built on the full simulation suite the authors plan to exploit in future work, potentially pushing accuracy below the per-cent level and extending to massive-neutrino cosmologies.
  • That n_eff evaluated at ν=1 suffices instead of a full ν-dependent shape implies the shape dependence acts mainly through the typical collapsing mass scale; this could be stress-tested on a cosmology with a strongly tilted power spectrum.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper calibrates a new semi-analytic halo mass function (HMF) fitting function within the Evolution Mapping framework. The multiplicity function of Eq. (11) extends a Tinker-like ansatz by including dependence on the effective spectral index n_eff and on x-tilde, an integrated growth-history variable defined in Eq. (8). The model is calibrated against the Aletheia and AletheiaMass N-body simulations for ten overdensity thresholds from Δ=150 to 1600, with careful attention to binning bias, cosmic variance, and resolution convergence. For the reference Δ=200 definition, the authors report per-cent-level residuals on the calibration suite, 3% residuals on the out-of-sample AletheiaEmu shapes, and roughly 5% residuals on Uchuu and on virial-overdensity catalogues. The paper additionally supplies interpolating formulae, Eq. (15), to make the model usable at intermediate overdensities, including the virial overdensity.

Significance. If the advertised accuracy survives scrutiny, this is a useful contribution: it extends Evolution Mapping from the power spectrum and velocity statistics to the HMF, provides a physically motivated alternative to redshift-dependent fitting functions, and ships a practical recipe for arbitrary overdensity thresholds. The paper has real strengths: the estimator in Eq. (13) avoids a known binning bias; the N_min convergence criteria are conservative; the AletheiaEmu and Uchuu comparisons are genuine out-of-sample tests; and the comparison with Tinker, Despali, and Euclid fitting functions is informative. The central Δ=200 calibration claim appears well supported. However, the advertised virial/interpolation component is not reproducible as written because part of Eq. (15) is undefined on the virial range used in the validation figures.

major comments (2)
  1. [§4.2, Eq. (15), and Figs. 8, 9, 11] Equation (15) contains A_n = 0.292 (y − 2.13)^0.67 and b_n = −0.86 (y − 2.0)^0.20. Over the virial range used in Figs. 8, 9, and 11, y = log10 Δvir is below 2.13 for essentially all displayed snapshots, and below 2.0 for the σ12=0.9 and 1.0 AletheiaMass snapshots (Δvir ≈ 94 and ≈ 88). Raising a negative base to a non-integer power is undefined in real arithmetic, so the plotted virial predictions cannot be reproduced from the stated recipe. This does not invalidate the Δ=200 calibration, but it directly undermines the paper's advertised interpolation/virial-accuracy claim. Please specify a real-valued continuation (or restrict the domain), report the values actually used to produce Figs. 8, 9, and 11, and re-evaluate the virial residuals after the fix. The systematic trends and reversed Aletheia ordering in Fig. 9 should be revisited in that light, since they may be artifacts of the unde
  2. [§5.1 and Fig. 10] The abstract and conclusions say the model 'maintains' per-cent-level accuracy on cosmologies with different power-spectrum shapes, but Fig. 10 shows residuals that are mostly within 3%, not 1%. This is a wording issue rather than a technical error, but the quantified claims in the abstract should match the plotted out-of-sample accuracy; otherwise readers may over-interpret the headline accuracy.
minor comments (5)
  1. [Fig. 9 caption] Typo: 'parameteter values' should be 'parameter values'.
  2. [Table 5 caption] Typo: 'formultiplicity' should be 'for the multiplicity'.
  3. [§3.2 and Appendix A] The meaning of STRICT_SO_MASSES is confusing. The main text says bound-only masses are obtained by setting STRICT_SO_MASSES=0, while Appendix A says STRICT_SO_MASSES=1 'prevents the code from excluding particles that are not gravitationally bound.' If both statements are literally true, the flag name is misleading; please clarify the configuration semantics.
  4. [Eq. (15)] The interpolation for log10 η is written as a function of Δ/1000, while the other coefficients are written in terms of y=log10 Δ. For consistency, express all terms in the same variable, and state the domain over which each power-law term is intended to be used.
  5. [Table 5 and §4.2] For Δ=1400 and 1600, the table reports only upper bounds on η (log10 η < −2.0 and < −2.4), while the text sets η=0 in this regime. The sentence recommending use of the tabulated fitted values for the ten calibrated thresholds should be reconciled with this fiat replacement.

Circularity Check

0 steps flagged

No significant circularity: the model is explicitly calibrated, x̃ is computed from growth history rather than from the HMF, and the out-of-sample AletheiaEmu/Uchuu tests are genuine predictions.

full rationale

The paper's derivation chain is a calibration, not an ab initio prediction. The nine parameters of Eq. (11) are fitted to N-body HMF measurements at ten overdensity thresholds (Section 4), and the paper states that η is treated as a free parameter: 'we do not assume a fixed memory scale. Instead, we will treat η as a free parameter in our model'. The variable x̃ of Eq. (8) is computed from Ωm(z)/f²(z) and the integrated growth history, not from the measured HMF; even though η is fitted, the non-universality variable itself is external to the HMF data. The Δ=200 claim is validated against AletheiaMass and Aletheia, while AletheiaEmu (Fig. 10) and Uchuu (Fig. 11) are held out, so those tests are genuine out-of-sample predictions. The self-citations (Sánchez et al. 2022, 2025; Esposito et al. 2024) are present and frame the 'physical motivation', but no load-bearing step reduces to an unverified prior result: x̃'s defining equation is given in the text, and the fit is scored directly against simulations. The paper itself flags the virial interpolation limitation ('This limitation likely explains the slight trend in Fig. 8'), which is an acknowledged accuracy caveat, not circularity. Separately, but not a circularity issue, Eq. (15) contains A_n = 0.292(y−2.13)^0.67 and b_n = −0.86(y−2.0)^0.20; for the Bryan & Norman virial overdensity Δvir ≈ 104 at z=0 (y ≈ 2.017), these powers have negative bases and are undefined in real arithmetic, so the advertised virial validation is not fully reproducible from the stated recipe. This is a correctness/reproducibility concern that does not affect the circularity score.

Axiom & Free-Parameter Ledger

10 free parameters · 7 axioms · 0 invented entities

The central claim is an empirical calibration, so the free parameters are the price of the fit. The out-of-sample tests (Uchuu, AletheiaEmu) provide independent support, reducing circularity burden. The main modeling axioms are the choice of the functional form of Eq. (11), the Gaussian-memory form of x-tilde, and the use of n_eff at ν=1.

free parameters (10)
  • A0 = 0.42 (Δ=150) to 0.89 (Δ=1600)
    Overall amplitude of the multiplicity function, fitted to N-body measurements at each Δ.
  • A_n = 0.040 to 0.294
    Coefficient of n_eff in the parameter A; fitted to the HMF data.
  • a_n = 0.212 to 0.234
    Coefficient of n_eff in the parameter a; fitted.
  • B0 = -0.6 (posterior peak; fixed to -0.6 in Table 5)
    Constant offset in the parameter B; listed as a free parameter in the fit, posterior peaked at -0.6.
  • B_x = 1.21 (Δ=150) to -0.06 (Δ=1600)
    Coefficient of x-tilde in B, controlling growth-history dependence; fitted.
  • B_n = 0.148 to -0.06
    Coefficient of n_eff in B; fitted.
  • b_n = -0.59 to -0.83
    Coefficient of n_eff in b; fitted.
  • C = 0.431 to 0.63
    Exponent in the exponential cutoff; fitted.
  • eta (log10) = -0.37 (Δ=150) to < -2.4 (Δ=1600)
    Width of the Gaussian memory kernel in Eq. (8); fitted with a log-uniform prior, only upper limits at high Δ.
  • Δ-interpolation coefficients = Various, Eq. (15)
    Eight empirical coefficients fitted to the Δ-dependence of the parameters, enabling interpolation to arbitrary overdensity thresholds.
axioms (7)
  • domain assumption Spherical collapse threshold δc = 1.686 (EdS value) is used for all cosmologies; its cosmology dependence is absorbed by other parameters.
    Section 2.1: 'We adopt the EdS value throughout, as the mild non-universality induced by the cosmology dependence of δc is effectively absorbed by the other parameters of our model.'
  • ad hoc to paper The multiplicity function form of Eq. (11) (Tinker-like with additive n_eff and x-tilde terms) is a valid universal ansatz across cosmologies and mass definitions.
    Section 2.4 presents this form without derivation; it is a model choice tested against simulations.
  • ad hoc to paper x-tilde defined in Eq. (8) with a one-sided Gaussian kernel G(τ'-τ|η) captures the relevant growth-history dependence, with η as the memory scale.
    Borrowed from Sánchez et al. (2025) for the power spectrum; here η is treated as free, but the kernel form is assumed.
  • domain assumption Evaluating the effective spectral index n_eff at ν=1 is sufficient to capture the shape dependence of the HMF.
    Section 2.3: 'we find that the non-universal behaviour of f(ν) is well captured by the effective slope evaluated at the characteristic peak height of ν=1'.
  • domain assumption Off-diagonal terms of the covariance matrix of the measured multiplicity function are negligible.
    Section 3.4: 'we set to zero the off-diagonal terms of the covariance of f(ν).'
  • domain assumption The linear halo bias of Tinker et al. (2010) accurately estimates cosmic variance errors on the HMF.
    Used in Eq. (14) to compute the sample variance contribution to the error bars.
  • domain assumption Paired-and-fixed initial conditions do not introduce biases beyond the quoted uncertainties.
    Section 3.1 states the use of paired and fixed initial conditions; Section 3.4 cites Villaescusa-Navarro et al. (2018) for the effect on HMF variance.

pith-pipeline@v1.3.0-alltime-deepseek · 23390 in / 12378 out tokens · 97595 ms · 2026-08-03T21:03:56.145993+00:00 · methodology

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read the original abstract

We present a new prescription for the halo mass function (HMF) built upon the Evolution Mapping framework. This approach provides a physical motivation to parametrise the non-universality of the HMF in terms of the recent history of structure formation and the local shape of the linear matter power spectrum. Our model was calibrated against measurements from N-body simulations, with halo samples defined by ten overdensity thresholds, $\Delta$, ranging from 150 to 1600 times the mean background matter density. For our reference mass definition, $\Delta=200$, the calibrated fitting function achieves per cent-level accuracy across a wide range of masses, redshifts, and structure formation histories, and maintains this performance when tested on cosmologies with different linear power spectrum shapes. This high level of accuracy is maintained across other mass definitions, degrading only slightly to the 5 per cent level at the highest values of $\Delta$. We also provide fitting formulae to interpolate the parameters as a function of $\Delta$, which allows for accurate modelling of HMFs defined by intermediate overdensities, with accuracy still well within 5 per cent when tested on halo catalogues defined by the virial overdensity threshold. Compared to other commonly used recipes, our prescription yields competitive or superior accuracy across all redshifts and cosmologies, successfully capturing the non-universal features of the HMF where other models exhibit systematic deviations. This work provides a high-precision modelling tool for cluster abundance analyses, and demonstrates the power of the evolution mapping framework for building accurate models of observables in the non-linear regime.

Figures

Figures reproduced from arXiv: 2511.16730 by Andrea Fiorilli, Andr\'es N. Ruiz, Ariel G. Sanchez, Matteo Esposito.

Figure 2
Figure 2. Figure 2: Upper panel: scaling of 𝑥 as a function of 𝜎12 in the Aletheia cosmologies defined in [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Stability of the measurements of the multiplicity function with respect to changes in the binning scheme, in two example snapshots of the AletheiaMass simulations. Assigning to each bin a representative mass value 𝑀(𝜈) evaluated at the central 𝜈 of the bin, as shown in the upper panels, can bias the estimate by a few per cent. Instead, summing masses directly into the estimator, as in equation (13), does n… view at source ↗
Figure 4
Figure 4. Figure 4: Upper panel: multiplicity function ˆ𝑓 (𝜈) measured in the Aletheia￾Mass simulations, corresponding to an overdensity threshold of Δ = 200. Lower panel: ratios of the measured values to the calibrated fitting function given in equation (11). To avoid crowding the plots with the measurements from all the AletheiaMass simulations, here and in subsequent figures we show, at each snapshot and for each mass bin,… view at source ↗
Figure 6
Figure 6. Figure 6: Ratios of ˆ𝑓 (𝜈) measured in the Aletheia and AletheiaMass simulations to the calibrated fitting function given in equation (11), for nine different values of the overdensity threshold Δ. The grey bands correspond to a 1 per cent difference. 4 CALIBRATION OF THE MODEL We begin by showing an example of the resulting ˆ𝑓 (𝜈) from the method described in Section 3 in the upper panel of [PITH_FULL_IMAGE:figure… view at source ↗
Figure 7
Figure 7. Figure 7: Interpolation of the best-fitting parameters as a function of the overdensity threshold. The black points represent the values listed in [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Ratios of the ˆ𝑓 (𝜈) defined in terms of the virial mass measured in the AletheiaMass simulations to the fitting function given in equation (11), using parameter values interpolated to the corresponding value of Δvir accord￾ing to equation (15). The grey band corresponds to a 1 per cent difference. particular, when fitting ˆ𝑓 (𝜈) of haloes defined with the two highest values of Δ, we only find an upper lim… view at source ↗
Figure 10
Figure 10. Figure 10: Ratios of ˆ𝑓 (𝜈) measured in a subset of the AletheiaEmu simula￾tions, with Δ = 200, to the calibrated fitting function given in equation (11). the dependence of the HMF on the growth history parameter, 𝑥˜. This limitation likely explains the slight trend in [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: Ratios of ˆ𝑓 (𝜈) from the Aletheia and AletheiaMass simulations, based on Δ = 200 (upper panels) and Δvir (lower panels), to other commonly used fitting functions (Tinker et al. 2008; Despali et al. 2016; Euclid Collaboration et al. 2023), and the one introduced in this work. The grey bands correspond to a 5 per cent difference. exhibit a spread comparable to that of the raw measurements shown in [PITH_F… view at source ↗

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Forward citations

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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  2. PHANTOM: A MATLAB and Octave Toolbox Connecting Linear Field Statistics to Dark Matter Halo Observables

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    PHANTOM is a public MATLAB/Octave toolbox for linear field statistics and halo observables in dark matter cosmology, validated to sub-percent agreement with Python packages colossus, hmf, and halomod.

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