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The Most Probable Outer Density Profile from Excursion Set Theory

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Removing the simplifying assumptions from the excursion-set derivation makes the predicted profile agree with the underlying distribution but diverge more from N-body simulations, so the universal-scaling profile is an effective fit…

desk verdict A solid and honest paper with a counterintuitive negative result, but the central comparison to N-body profiles borrows conventions from prior work and needs to be made rigorous before the quantitative divergence claim is taken at face value. read the letter →

arxiv 2608.13347 v1 pith:XVJG66D2 submitted 2026-08-13 astro-ph.CO

Ericka Florio , Vasiliki Pavlidou This is my paper · ORCID
classification astro-ph.CO
keywords outerdensityprofileexcursionsettheorydoubledistributionturnaroundradiusuniversalscalingwindowfunctionMarkovianrandomwalksphericalcollapse
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

The paper asks whether the 'universal-scaling' outer density profile of galaxy clusters—a formula derived from excursion set theory under a stack of simplifying assumptions—is still obtained when those assumptions are removed one by one. The authors re-derive the most probable profile from the double distribution, the joint probability of a collapsed object's mass and the density contrast of its surroundings, this time transforming properly to nonlinear density before taking the mode, keeping the mass-dependent term, and using a realistic matter variance. They find that each correction makes the analytic mode agree better with the numerically evaluated distribution but worse with the N-body simulation profiles that motivated the original formula. They attribute the growing mismatch to the window function: excursion set theory's sharp filter in Fourier space (k-space) produces independent, Markovian increments, while simulated profiles are built with a top-hat filter in real space whose increments are correlated. The paper concludes that the universal-scaling profile should be treated as a few-parameter effective description fitted to simulations, not as a first-principles prediction of excursion set theory.

What carries the argument

The load-bearing object is the double distribution (DD), the joint probability for a collapsed object of mass $m$ to be embedded in a region of overdensity at the larger scale $\beta m$, and in particular the mode of slices of that distribution, which is interpreted as the most probable outer density profile. The paper's technical core is a cubic polynomial, $2A'\tilde\delta_c X^3 + (2A'\eta - 2B'\tilde\delta_c)X^2 - (2B'\eta + 2\tilde\delta_c + \tilde\delta_c^2)X - \eta(1+\tilde\delta_c)=0$ with $X=\hat\rho^{-1/\tilde\delta_c}$, that gives the mode after the distribution has been properly transformed from linear to nonlinear overdensity; it replaces the earlier shortcut of differentiating first and converting afterwards. Around this core, the machinery includes the choice of spherical-collapse conversion, the Bardeen transfer function for the mass variance $S(m)$, and the sharp k-space window whose independent increments make the density trajectory Markovian—the property that the paper identifies as the source of the mismatch with top-hat-filtered simulation profiles.

What would settle it

Re-extract the most probable outer density profile from the same N-body simulations using a filter whose Fourier increments are independent (or a smooth k-space filter matched to the analytic variance convention), holding mass definitions fixed, and check whether the corrected analytic modes now match; if the divergence persists under matched windows, the window-function explanation would be refuted.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is negative: the earlier simplifying assumptions were not innocent shortcuts. When the double distribution is transformed to nonlinear density before the mode is taken, when the dropped term that carries mass dependence is retained, and when the power-law variance is replaced by a Bardeen transfer-function variance, the resulting most probable profile continues to match the numerical mode of the double distribution—all internal consistency checks pass—but it departs from the N-body profiles, and the departure grows as each assumption is relaxed. The paper reads this as evidence that the discrepancy lies inside excursion set theory itself, in the Markovian random walk induced by the sharp k-space window function, which is not the window used to measure simulated profiles. It therefore concludes that the universal-scaling profile is not a first-principles prediction of the outer density profile; it is a functional form that works well as an effective description once its few parameters are fitted to simulated data.

Load-bearing premise

The divergence verdict assumes that the published N-body profiles, extracted with a top-hat window and particular mass definitions, are directly comparable to the sharp-k analytic calculation; if the two window conventions are not commensurable, the measured divergence is distorted.

Editorial extensions

If this is right

  • The turnaround-radius probe of dark energy remains usable, but its theoretical profile must be calibrated against N-body simulations for each mass range rather than taken as a parameter-free prediction.
  • A correct first-principles prediction of the outer density profile will require a non-Markovian formulation of excursion set theory with correlated density trajectories.
  • The universal-scaling profile's empirical success means its functional form and fitted parameters still carry information about the quasi-linear density field, even though its excursion-set derivation is not first-principles.
  • The correctly transformed mode of the double distribution can serve as an internal benchmark for numerical implementations of excursion set theory, independently of its agreement with simulations.

Reading between the lines

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

  • Editorial extension: if the window-function diagnosis is correct, then re-extracting simulated profiles with a sharp-k-like filter that has independent Fourier increments should restore agreement with the corrected analytic modes; the paper does not run this test.
  • Editorial extension: the monotonic growth of the divergence suggests the simplifying assumptions and the Markovian approximation partially cancel each other's errors in the original profile, so fitting the universal-scaling profile to simulations may be absorbing both effects at once.
  • Editorial extension: because the full spherical-collapse conversion could not be evaluated for $\hat\rho \lesssim 14$, the behavior of the correctly transformed mode in exactly the low-density regime where the main conclusions are drawn still rests on the approximate conversion.
  • Editorial extension: the same Markovian-versus-correlated mechanism, if real, should affect other excursion-set predictions such as conditional mass functions and clustering statistics, not only outer density profiles.
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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 re-derives the most probable outer density profile from excursion set theory's double distribution (DD). It transforms the DD from linear overdensity to nonlinear matter density, derives a cubic polynomial for the mode of the transformed distribution (Eq. 6), and evaluates the mode numerically across mass and clustering parameter β. The paper then compares the resulting analytic profiles with N-body profiles taken from Korkidis & Pavlidou (2024, Fig. 5), finding that the analytic mode agrees with the numerically realized DD but diverges more strongly from the simulated profiles as simplifying assumptions are relaxed. The authors attribute this mismatch to the difference between the sharp k-space window function used in excursion set theory and the configuration-space top-hat filter used in simulations, and conclude that the universal-scaling (US) profile should be used only as a few-parameter effective description fitted to simulations.

Significance. If the negative result is established, it is a useful clarification for the field: the US profile is not a first-principles prediction of excursion set theory, and its success as a fit to simulations must be understood through the window-function/Markovian assumption. The paper's internal derivation is self-contained, the analytic mode is validated against a direct numerical evaluation of the DD (Figs. 2 and 5), and the analysis code is released, which are concrete strengths. However, the central comparison to N-body profiles is not controlled: the analytic curves are functions of β while the simulated profiles are functions of radius, and the β↔radius mapping is neither derived nor tested. The conclusion that the divergence grows as assumptions are relaxed is therefore not yet fully supported, and in addition the full spherical-collapse conversion could not be evaluated in the mode region, so assumption (3) was not actually relaxed in the main comparison.

major comments (2)
  1. [Sec. 3, Fig. 6] The central claim that the corrected profile diverges from N-body profiles, and that this divergence grows as assumptions are relaxed, rests on an uncontrolled comparison: the analytic modes in Fig. 6 are computed as functions of the clustering parameter β (via Eqs. 6 and 17), while the simulated profiles are imported from Korkidis & Pavlidou (2024, Fig. 5) as functions of radius. The mapping from β to radius is not derived, stated, or tested anywhere in this paper. Without such a mapping, the horizontal offset between the analytic curves and the simulated profiles is uncontrolled, so the apparent growth of the divergence could be an artifact of mismatched mass, radius, or window conventions rather than a physical effect. Please provide the β↔radius mapping used to place the modes in Fig. 6, or redo the comparison with simulations analyzed under the same mass and window definitions as the analytic calculation.
  2. [Sec. 2.3 and Fig. 4] The full spherical-collapse conversion, Eq. (8), could be evaluated only for ρ̂ ≳ 14 because of the stiffness of Eq. (11), while all modes discussed in Sec. 3 lie at ρ̂ < 10. Consequently, the 'relaxed' profiles shown in Fig. 6 still use the approximate conversion, Eq. (7), in exactly the region where the modes occur. The paper therefore does not actually relax assumption (3) in the comparison that supports the successive-relaxation claim. This limitation is acknowledged in the text, but it should be made explicit that the 'divergence grows as we relax successive assumptions' statement applies only to assumptions (1), (2), (4), and (5), and not to assumption (3), unless the full conversion is evaluated down to the mode region.
minor comments (5)
  1. [Eq. (4)] Equation (4) is typeset ambiguously as ρ̂_us = (1 − β − γ)^(−δc); it should be ρ̂_us = (1 − β^{−γ})^{−δc} (or an equivalent explicit form), since the power-law index γ enters as an exponent of β.
  2. [Eq. (6) and Figs. 5–6] Equation (6) is a cubic polynomial in X, so it has up to three real roots; the manuscript does not state which root corresponds to the physical mode nor how that branch is selected in Figs. 5 and 6. This should be specified to make the mode calculation reproducible.
  3. [Sec. 3] The sentence in Sec. 3 that says the results 'rule out assumptions (2)–(4)' is inaccurate, because assumption (3), the spherical-collapse approximation, is not evaluated in the mode region; only assumptions (2), (4), and (5) are quantitatively addressed. This wording should be corrected.
  4. [Fig. 5 caption] The Fig. 5 caption appears to have a truncated axis label; the horizontal axis should be explicitly labeled as β (and the vertical axis as the mode ρ̂) so that the figure is interpretable without reference to the main text.
  5. [References] The reference to Tanoglidis et al. (2016) is listed as an arXiv identifier only; if the paper has appeared in a journal, the full bibliographic information should be provided.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the corrected-profile derivation is parameter-free, and the divergence claim is an external test against N-body data.

full rationale

The derivation of the corrected mode (Eq. 6) is self-contained: it starts from the double distribution (Eq. 1), applies the transformation Eq. (5), and solves for the mode with fixed cosmological parameters and the Bardeen transfer function (Eq. 15); no parameter is fitted to N-body data to produce the corrected profile. The agreement between the analytic root of Eq. (6) and the numerical mode of the same DD (Figs. 2, 3, 5) is an internal consistency check of the calculus and code, not a physical prediction, and the paper presents it as such. The central physical claim—that the relaxed profiles diverge from N-body profiles increasingly—is an external comparison to simulation profiles taken from Korkidis & Pavlidou (2024); those profiles are data, not an input of the derivation, so the comparison cannot reduce to the derivation by construction. Self-citations to Pavlidou & Fields (2005) and Korkidis & Pavlidou (2024, 2025) are numerous, but the load-bearing steps (solving the transformed-mode cubic, evaluating S_B(m) from Bardeen, numerically resolving the DD) are performed in this paper rather than merely imported. The admitted limitations—full spherical-collapse conversion not evaluable below ρ̂≈14 so Eq. (7) is retained in the mode region, and the possible mismatch of β-axis versus radius in the N-body comparison—are correctness risks about uncontrolled comparison, not instances where a result is equivalent to its input. No circular step can be exhibited.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the excursion-set-theory framework, the spherical collapse mapping, and the previously published N-body profiles used for comparison. The new derivation introduces no new entities and fits no new parameters; the only fitted quantity in the comparison is the slope gamma of the US profile, which the paper concludes must be treated as an effective parameter.

free parameters (2)
  • gamma (power-law slope of S(m) in the US profile) = fit to each mass group of simulated clusters in Korkidis & Pavlidou (2024)
    The universal-scaling profile used for comparison in Fig. 6 depends on this slope fitted to simulated cluster groups; the paper concludes that such fitting is what makes the US profile an effective description rather than a prediction.
  • beta (clustering parameter) = scanned over 1.25 to 2.25
    beta parameterizes the mass ratio between the collapsed object and its environment; the paper computes profiles over a range of values rather than fitting it to data.
assumptions (5)
  • domain assumption Excursion set theory with a sharp k-space window function yields a Markovian random walk for the overdensity
    Central to the double distribution derivation in Section 2.1 and to the paper's diagnosis in Section 4 that the mismatch stems from the Markovian property.
  • domain assumption The double distribution of Pavlidou & Fields (2005), Eq. (1), correctly describes the joint distribution of collapsed mass and extrapolated overdensity
    Used as the starting point for the mode derivation in Section 2.1; the paper does not re-derive or test this distribution itself.
  • domain assumption The spherical collapse model maps linear to non-linear overdensity, and the approximate conversion Eq. (7) is accurate to a few percent in the relevant range
    Used throughout Appendix A and Section 2.3; the authors could not evaluate the full conversion near the mode (rho-hat < 10), so the transformed profile depends on this approximation.
  • standard math The Bardeen transfer function and the sharp-k relationship k(m)=(6 pi^2 rho_m,0/m)^(1/3) give the correct matter variance S(m)
    Assumed in Section 2.4 for computing S_B(m); this is a standard cosmological input.
  • domain assumption The N-body profiles from Korkidis & Pavlidou (2024) accurately represent the outer density profiles of simulated Lambda-CDM clusters
    The paper's comparison in Section 3 and Fig. 6 relies on these published profiles without re-analyzing the simulations.

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Pith. "Pith review of The Most Probable Outer Density Profile from Excursion Set Theory." pith.science (2026). https://pith.science/paper/XVJG66D2

@misc{pith2026260813347,
  author       = {Pith},
  title        = {Pith review of: The Most Probable Outer Density Profile from Excursion Set Theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XVJG66D2}},
  note         = {Machine review of arXiv:2608.13347}
}
read the original abstract

Measurements of the turnaround radius around galaxy clusters can be used to break the degeneracy between measurements of the present day energy density of matter and dark energy. Korkidis & Pavlidou showed that the turnaround radius coincides with the first point of deviation between outer density profiles in N-body simulations and the analytic profile predicted by excursion set theory. However, their analytic profile relied on a number of simplifying assumptions, which may each introduce systematic error. We evaluate the effect of these assumptions on the shape of the analytic profile and its correspondence with simulated outer density profiles. We relax the key simplifying assumptions and re-derive the mode of the outer density profile from excursion set theory. We then numerically resolve the double distribution (DD) across a range of masses and clustering parameters, and compare the numerical mode estimate to the re-derived analytic profile. We find excellent agreement between our analytic profile and the numerically-realized DD. However, our analytic profiles diverge from N-body profiles, and this divergence grows as we relax successive assumptions. We relate this mismatch to the differing window functions used in the analytic and simulation-based approaches, which respectively yield Markovian and correlated density trajectories. We conclude that the analytic profile proposed by Korkidis & Pavlidou should only be used as a few-parameter effective description of the most-probable outer density profile, with parameters fitted to results of cosmological simulations.

Figures

Figures reproduced from arXiv: 2608.13347 by the authors.

Figure 2
Figure 2. Conditional slices of the joint distribution at m = 1.3 × 1014M⊙ (blue) and 1.7×1015M⊙ (orange). Dashed lines show the untransformed double distribution, and solid lines show the transformed double distri￾bution. Red dots indicate the analytic mode of each curve, and blue stars indicate the numeric mode of each curve. analytic solution is both mathematically correct, and is properly resolved by our python code. In … view at source ↗
Figure 3
Figure 3. The transformed double distribution, sliced at m = 1.3 × 1014M⊙ (blue) and m = 1.7 × 1015M⊙ (orange/red). The solid lines show the transformed distribution without the structure-in-structure term and with a power law approximation assumed. The crosses in panel (a) show the transformed double distribution with the structure-in-structure term included. The dashed lines in panel (b) show the double distribution where t… view at source ↗
Figure 4
Figure 4. The transformation ˆρ → δ˜ l , as given by the full solution to the non-linear evolution equation, Eq. (8) (green), and the approximate equation Eq. (7) (purple). Panel (a) shows the conversion itself, and panel (b) shows the derivative of this conversion. values of β between (1.25, 2.25) [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figures from the paper (2 more)
Figure 5
Figure 5. Figure 5: The most probable density profile for the transformed distribu￾tion at m = 1.3 (blue), 5 (green), and 17 (red) times 1014M⊙. The solid line shows the analytic mode, and the crosses show the numeric mode [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: A comparison between both the transformed (dotted) and un￾transformed (dot-dashed) double distributions’ analytic modes, and the results from Korkidis & Pavlidou (2024), showing the profile extracted from N-body simulations (solid) alongside the universal scaling profi…

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