The Density Profile of Dynamical Halos

Aakanksha Adya; Calvin Osinga; Edgar M. Salazar; Eduardo Rozo; Tristen Shields; Ze'ev Vladimir

REVIEW 4 major objections 5 minor 28 references

The Density Profile of Dynamical Halos

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper claims that individual dynamical halos of fixed mass have orbiting density profiles controlled by a single variable, the halo radius $r_{\rm h}$, with halo radii scattered by about 16% at fixed mass and by about 11% after…

desk verdict Plausible but unproven: the claim that individual dynamical halo profiles have one degree of freedom needs independent validation and covariance handling. read the letter →

arxiv 2507.00410 v2 pith:56HNOAO3 submitted 2025-07-01 astro-ph.CO

Tristen Shields , Edgar M. Salazar , Eduardo Rozo , Aakanksha Adya , Calvin Osinga , Ze'ev Vladimir This is my paper · ORCID

classification astro-ph.CO

keywords darkmatterhalosdensityprofilesdynamicalorbitingparticleshaloradiusformationtimecosmologicalsimulationsmassscatter

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

This paper adopts the dynamical-halo definition in which a halo is the collection of orbiting particles around its self-generated potential, and asks how the orbiting density profile of such a halo depends on mass. It claims that at fixed mass the profile of an individual halo is controlled by a single dynamical variable, the halo radius $r_{\rm h}$, because the fitted inner slope $\alpha_\infty$ is tied to $r_{\rm h}$ through $\Delta\alpha_\infty = 0.05 + \ln R$. The halo radius is lognormally scattered by about 16% at fixed mass; including the relative formation time $a_{\rm RF}$ through $r_{\rm h,mod} = (1.91-a_{\rm RF})\,r_{\rm h,st}(M_{\rm orb})$ lowers the scatter to about 11%. If correct, a halo's mass and formation time can be read off from the shape of its orbiting profile alone.

What carries the argument

The load-bearing object is the halo radius $r_{\rm h}$, the length scale at which the exponential truncation $\exp(-r^2/2r_{\rm h}^2)$ turns on in the adopted profile model [9]. With the inner scaling $\epsilon$ fixed at 0.037, the slope $\alpha_\infty$ is replaced by the deterministic relation $\Delta\alpha_\infty = 0.05 + \ln R$, where $R \equiv r_{\rm h}/r_{\rm h,st}(M_{\rm orb})$, turning the two-parameter fit into a one-parameter fit. Then $r_{\rm h}$ is linked to the relative formation time $a_{\rm RF} = a_{60}/{\rm median}(a_{60}|M_{\rm orb})$ by $r_{\rm h,mod} = (1.91-a_{\rm RF})\,r_{\rm h,st}(M_{\rm orb})$, which tightens the predicted radius. The slope–radius relation is what makes the profile a single-degree-of-freedom family; the formation-time relation is what makes the radius partly predictable from accretion history.

What would settle it

Fit every halo in an independent simulation with both $\alpha_\infty$ and $r_{\rm h}$ free, then refit with $\alpha_\infty$ forced by Equation (12); if the forced fits degrade significantly, or if the free fits scatter about $\Delta\alpha_\infty = 0.05 + \ln R$ much more widely than the quoted errors, the single-degree-of-freedom claim is not supported.

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Extended reading notes

Core claim

The central discovery is that the orbiting profiles of dynamical halos form an effectively one-parameter family at fixed mass. Fitting each halo with the truncated power-law model $\rho_{\rm orb}(x) = A(x/\epsilon)^{-\alpha(x)}\exp(-x^2/2)$ yields two shape parameters, the asymptotic slope $\alpha_\infty$ and the halo radius $r_{\rm h}$; the paper shows these are tightly correlated within a mass bin, $\Delta\alpha_\infty = 0.05 + \ln(r_{\rm h}/r_{\rm h,st}(M_{\rm orb}))$. This correlation lets every halo's profile be described by $r_{\rm h}$ alone, with the slope imposed by Equation (12). The halo radius at fixed orbiting mass is lognormally distributed with intrinsic scatter $\sigma_{\ln r_{\rm h}|M_{\rm orb}} = 0.16$; late-forming halos are more compact, and using the relative formation time through Equation (18) reduces the scatter to $\sigma_{\ln r_{\rm h}|M_{\rm orb},a_{\rm RF}} = 0.11$. The paper takes these two scalings as evidence that the orbiting profile of a dynamical halo is a single-degree-of-freedom object whose extent encodes mass and formation history.

Load-bearing premise

The central assumption is that the tight correlation between fitted inner slope and halo radius, $\Delta\alpha_\infty = 0.05 + \ln R$, reflects a physical single degree of freedom and not a degeneracy in the two-parameter fitting procedure, because the same fits that define the relation are then refit under it.

Editorial extensions

If this is right

A halo's orbiting density profile at fixed mass is fully characterized by one number, $r_{\rm h}$, so the profile shape carries no independent information beyond its spatial extent.
From a measured profile slope and radius, Equations (11) and (18) can be inverted to estimate both the orbiting mass and the relative formation time of the halo.
The intrinsic scatter in $r_{\rm h}$ at fixed mass is about 16%; adding formation time reduces it to about 11%, setting the accuracy with which mass, and mass plus formation time, predict the profile extent.
Late-forming halos are more compact and have shallower inner slopes, extending the known concentration–formation-time connection to the orbiting profiles of dynamical halos.
The same trends appear with an alternative four-parameter profile parameterization, suggesting the single-degree-of-freedom behavior is not an artifact of the adopted fitting function alone.

Reading between the lines

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

If the relation $\Delta\alpha_\infty = 0.05 + \ln R$ holds across redshifts and simulation resolutions, the orbiting profile could serve as a low-cost estimator of halo mass and formation time that bypasses full merger-tree construction.
Because the residual scatter after including formation time is still 11%, a natural next test is whether combining $a_{\rm RF}$ with the recent accretion rate removes more of the scatter than either variable alone; the paper compares but does not combine the two.
If profile shape determines mass and formation time, then lensing or satellite-kinematics measurements of the inner slope may be able to infer halo radius, turning the single-degree-of-freedom claim into a route from observed profiles to halo assembly history.
The 16% scatter in halo radius at fixed mass implies a corresponding spread in the truncation scale of the orbiting profile, which should be propagated into stacked halo-model predictions of large-scale structure.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

First, what you should know: this is a solid empirical paper with public code and transparent fitting methods, but the headline claim—that individual dynamical halo profiles at fixed mass live on a one-parameter family—is not established as strongly as the paper suggests. It rests on a correlation measured from the same fits that define both variables, with no independent simulation validation and no error bars on the headline scatter values.

What's new: the observation that the two fitted profile parameters (inner slope and halo radius) are tightly correlated at fixed mass, and that the scatter in halo radius is ~16%, shrinking to ~11% when relative formation time is included. The comparison to Diemer's recent paper is honest and useful, and the code availability is a plus. The internal consistency check in Section IV is a nice touch, even if it's approximate.

Now the soft spots, in proportion. The biggest is the single-degree-of-freedom claim. The authors fit each halo with two free parameters, plot the resulting alpha_inf against r_h, fit a line to that plot, then impose the line and refit everything with only r_h free. That procedure cannot validate the relation—it assumes it. They never quantify the scatter around the relation, and they don't propagate the covariance between the two fitted parameters. The stress-test concern about a fitting degeneracy (Poisson noise shifting r_h and alpha_inf together) is legitimate. It's not fatal, because Diemer's Figure 10 shows a similar correlation using a different profile model and different simulations, suggesting some degree of physical reality. But the paper doesn't lean on that as external validation; it just cites it as qualitative agreement. A referee should ask for a mock-catalog test or an independent simulation fit.

Second, the headline scatter numbers (16% and 11%) have no uncertainties. The variance model includes a particle-noise term, but the error on the intrinsic scatter is never quoted. For a paper whose main quantitative results are these two percentages, that's a real gap.

Third, the formation-time analysis is restricted to halos with a60 > 0.4 and orbital masses above 10^14 h^-1 M_sun, so the 11% figure applies to a limited range. The paper acknowledges this, but it should be made clear that the low scatter claim is conditional.

Who's this for: anyone modeling halo density profiles or using cluster profiles for mass calibration. It's a useful empirical contribution, but I'd treat the one-parameter compression as provisional until validated elsewhere.

My recommendation: send it to a serious referee. The question is worth asking, the execution is mostly careful, and the issues are addressable in revision. A competent referee will likely ask for cross-validation and error propagation, but that doesn't make the paper unworthy of review.

Referee Report

4 major / 5 minor

Summary. The paper analyzes the dark matter density profiles of dynamical halos in a single N-body simulation. It fits individual halo profiles with the two-parameter model of Salazar et al., identifies a tight correlation between the fitted slope offset Δα∞ and the scaled halo radius at fixed mass, adopts this correlation as a one-parameter model, measures the scatter in halo radius, and studies its dependence on formation time and accretion rate. The central claims are that individual profiles at fixed mass have one effective degree of freedom, the halo radius, and that the profile shape can be used to read off both mass and formation time.

Significance. If the single-degree-of-freedom claim is established, this would be a useful empirical simplification of halo structure: the orbiting density profile of dynamical halos would be a one-parameter family at fixed mass, with a modest additional dependence on formation time. The paper is transparent about its methods, makes the analysis code publicly available, and includes a useful comparison with the independent Diemer parameterization. The main limitation is that the central correlation is calibrated and then imposed on the same data, so the paper currently falls short of demonstrating that the correlation is physical rather than a fitting degeneracy.

major comments (4)

[III.B (Eq. 11), Fig. 2] The central claim that individual profiles have a single effective degree of freedom rests entirely on the tight Δα∞–ln R relation, but this relation is measured from the same two-parameter fits that define both variables, and the subsequent one-parameter refits in Section III.C impose Eq. (11) and therefore cannot validate it. The fixed value ε=0.037 places the slope transition at x∼0.037, below the fitted radial range r>0.09 h−1 Mpc, so the data constrain only the approach of α(x) to α∞ over the fitted interval; this is exactly the regime in which a degeneracy between rh and α∞ could masquerade as a physical correlation. I ask the authors to demonstrate, for example by fitting mock halos with independent rh and α∞, by cross-validating Eq. (11) on an independent subsample or simulation, or by reporting the individual-fit covariance and showing that its noise direction is not aligned with Eq. (11), that the correlation is not an artifact of the fitting procedure.
[III.C (Eq. 14)] The intrinsic scatter σln rh|Morb = 0.16 is derived by fitting Var(ln rh/rh,st|N) = σ0^2 + k/N to the one-parameter refits, but the fit is not shown, no uncertainty on σ0 is quoted, and the variance model is not tested. Since these refits already assume Eq. (11), any systematic error in the assumed relation enters the quoted scatter. Please report the fitted k, the uncertainty on σ0, and validate the particle-noise decomposition with mocks or by checking stability across radial binning and the choice of δ in Eq. (5).
[IV (Eqs. 17-18)] The formation-time relation is calibrated on the same high-mass sample used to define aRF via median(a60|Morb), and the result is summarized by the residual scatter σ = 0.11 without an uncertainty or residual diagnostics. The paper should show that the residuals are consistent with zero mean and constant variance as a function of mass and aRF, and quote an uncertainty on the 0.11 value. The abstract's wording that only a small fraction of the scatter is due to formation time should also be quantified relative to the variance: (0.16)^2 versus (0.11)^2 leaves roughly half the variance unexplained, which is not obviously a small fraction.
[IV (Eq. 22)] The internal consistency check between Eq. (20) and Eq. (21) yields a 0.029 difference between the right-hand side (−0.094) and the directly measured value (−0.064). The attribution of this offset to non-Gaussian tails is plausible, but it should be verified, for instance by showing that the tail correction has the required sign and size. As it stands, the model has a small unmodeled offset in a relation that is used to support the formation-time dependence of the halo radius.

minor comments (5)

[VI, bullet list] The second bullet says that 'at fixed mass, the slope of the profile (α∞) is tightly correlated with halo mass'; this appears to be a typo, as the text and Fig. 2 show a correlation with halo radius (or scaled radius R), not with halo mass.
[VI, first comparison paragraph] The phrase 'characterized the orbiting orbiting profiles of spherical halos' contains a duplicated word and should be corrected.
[Throughout] The manuscript mixes the spellings 'halos' and 'haloes'; please choose one and use it consistently.
[III.C (Eq. 14)] The best-fit value of k in Eq. (14) is not reported, which makes the variance model difficult to reproduce; please add the fitted value and its uncertainty.
[III.B (Eq. 10)] The uncertainties on the fitted line come from jackknifing the box, but the number of jackknife realizations and the scatter around the line are not stated; reporting these would help readers assess the tightness of the correlation in Fig. 2.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central claim is an empirical correlation between independently fitted profile parameters, not a definitional identity or a fitted input renamed as a prediction.

full rationale

This paper is an empirical characterization rather than a derivation, and no step reduces by construction to its own inputs. The key relation Δα∞ = 0.05 + ln R (Eq. 11) is fit to the same individual two-parameter fits used to define Δα∞ and R, but those two parameters are independently fitted degrees of freedom in Eq. (1); neither is defined in terms of the other. The one-parameter refits in Sec. III.C impose Eq. (12), so they do not independently validate the relation, but the paper does not present those refits as the validation; the inference of a single effective degree of freedom rests on the small scatter about the fitted correlation in Fig. 2. The quoted scatters σln rh|Morb = 0.16 and σln rh|Morb,aRF = 0.11 are measured residual widths, not quantities forced to be small by the calibration, and the formation-time relation Eq. (18) is fit before the residual scatter is evaluated. The Salazar et al. [9] parameterization is adopted from prior work with overlapping authors, but it is an input model, not the target result, and the qualitative α∞–rh correlation is cross-checked against Diemer's [17] independent analysis. The concern that the correlation might reflect a fitting degeneracy rather than an intrinsic degree of freedom is a robustness/correctness question requiring covariance or mock tests; it is not a circularity of the paper's logic.

Assumptions & free parameters 10 free parameters · 6 assumptions · 0 invented entities

No new physical entities are introduced. The paper's load-bearing content is a set of empirically fitted constants, some from this paper and some from the authors' earlier stacked-profile work, and the central single-degree-of-freedom claim depends on these fits without full propagation of their uncertainties.

free parameters (10)

epsilon = 0.037
Inner slope transition scale fixed from the stacked-profile fit of Salazar et al. 2025 and adopted without refit for individual halos.
Stacked mass-scaling parameters (rh,p, rh,s, aInf,p, aInf,s) = 840.3 h^-1 kpc, 0.226, 2.018, -0.050
Adopted from Salazar et al. to define the stacked normalizations rh,st and aInf,st used in R and Delta-aInf.
Delta-aInf intercept = 0.05
Intercept of the aInf-ln R relation in Eq. 11, fit to the individual halo fits and adopted as the fiducial value.
Delta-aInf slope = 1.0
Slope of the aInf-ln R relation, measured as 1.002 +/- 0.021 and rounded to 1.0.
Mean log-radius offset = -0.064 (ratio 0.937)
Peak and mean of the log-radius distribution at fixed mass; all mass bins peak at rh/rh,st = 0.937.
Intrinsic scatter at fixed mass = 0.16
sigma_ln rh given Morb after subtracting a particle-noise term k/N; this is the central distribution-width result.
Formation-time relation intercept a0 = 1.91
Intercept in R = a0 + sa aRF, rounded from the measured 1.89 +/- 0.03.
Formation-time relation slope sa = -1.0
Measured as -0.98 +/- 0.02 and adopted as -1.0; relates halo radius to relative formation time.
Residual scatter at fixed mass and formation time = 0.11
Intrinsic scatter in ln rh after applying Eq. 18; this is the primary reduction claim.
Cost-function percent error delta = 0.05
Hand-chosen 5% error floor in the profile-fitting cost function; the authors state results are only marginally sensitive to it.

assumptions (6)

domain assumption The Quijote simulation faithfully represents CDM halo structure at the resolved scales used.
All results come from one N-body simulation with a single cosmology; no convergence study or independent cosmology check is presented.
domain assumption The Garcia et al. orbiting/infalling classification correctly separates particles and produces accurate accretion times.
The dynamical halo definition and aacc tagging from Ref. [1] are taken as given; the paper does not test the classifier's systematic errors.
domain assumption The Salazar et al. profile model with fixed epsilon = 0.037 is a valid description of individual halo orbiting profiles.
The parameterization was calibrated on stacked profiles; applying it to individual halos and reading physical meaning from its internal correlations is not independently validated.
ad hoc to paper The CDF level a60 = 0.60 and its median normalization define a useful formation-time proxy.
a60 was chosen 'upon inspection' to separate CDF curves; different choices could change the inferred formation-time dependence.
ad hoc to paper The fitted aInf-ln R relation (Eq. 11) is exact enough to impose in one-parameter refits.
Adopted as fiducial and used to reduce the model to one degree of freedom; scatter in this relation is not propagated into the one-parameter fits.
domain assumption Rockstar/ConsistentTrees halo masses and accretion rates are accurate for the comparison in Figure 5.
The accretion-rate comparison relies on merger tree quantities for spherical-overdensity halos, while the authors note they could not self-consistently estimate accretion rates for dynamical halos.

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Cite this review

Pith. "Pith review of The Density Profile of Dynamical Halos." pith.science (2026). https://pith.science/paper/56HNOAO3

@misc{pith2026250700410,
  author       = {Pith},
  title        = {Pith review of: The Density Profile of Dynamical Halos},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/56HNOAO3}},
  note         = {Machine review of arXiv:2507.00410}
}

abstract

Among the most fundamental properties of a dark matter halo is its density profile. Motivated by the recent proposal by Garc\'ia et al. [R. Garc\'ia et. al., MNRAS 521, 2464 (2023)] to define a dynamical halo as the collection of orbiting particles in a gravitationally bound structure, we characterize the mean and scatter of the orbiting profile of dynamical halos as a function of their orbiting mass. We demonstrate that the orbiting profile of individual halos at fixed mass depends on a single dynamical variable -- the halo radius $r_{\rm h}$ -- which characterizes the spatial extent of the profile. The scatter in halo radius at fixed orbiting mass is $\approx 16\%$. Only a small fraction of this scatter arises due to differences in halo formation time, with late-forming halos being more compact (smaller halo radii). Accounting for this additional correlation results in an $\approx 11\%$ scatter in halo radius at fixed mass and halo formation time.

Figures

Figures reproduced from arXiv: 2507.00410 by the authors.

**Figure 1.** Left: Orbiting density profiles (r 3 ρorb vs. r) of 300 random halos in three mass bins as labeled. Each mass bin has 100 randomly selected halos. Also shown as points with error bars are the mean orbiting profiles from all halos in the bin. The solid black lines are the best fit model obtained using Equation 1. Scales r ≤ 0.09 h −1 Mpc (gray band) are subject to force-softening, and are therefore not fit. Top-Right… view at source ↗

**Figure 2.** The color scale shows the number of dark matter [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 4.** After binning all of the halos by Morb, then binning again by ln R, we take CDFs of the particle accretion times of all the particles belonging to halos in those bins. A horizontal line is drawn when the CDF reaches 0.6, as at this point there is visually a clear distinction between most of the ln R bins that is approximately constant with mass. with a60 < 0.4 is not resolved. For this reason, for the remainder of t… view at source ↗

Figures from the paper (3 more)

**Figure 5.** Figure 5: Left: Correlation between the normalized halo radius (Equation 9) and a halo’s relative formation time (Equation 16) at fixed mass, for halos of mass Morb > 1014h −1 M⊙. Our best fit line for this correlation is shown in red. Right: Correlation between halo radius at f…

**Figure 3.** Figure 3: That is, the reduction in parameters space in [PITH_FULL_IMAGE:figures/full_fig_p006_3.png]

**Figure 6.** Figure 6: Visual representation of the relationships between [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Diemer focuses on the properties of ρorb for spheri- cal overdensity halos, whereas we focus specifically on dynamical halos. Even though the difference in algorithms has a minimal impact on the orbit- ing profiles of individual halos [18], the choice of definition matters with regards to the scatter and correlation of halo properties with mass

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Salazar et al

in setting ϵ = 0.037, leaving only two free parameters with which to fit the orbiting profile of dynamical halos. Salazar et al. [9] fit the stacked density profile of dynam- ical halos, finding both α∞ and rh vary with mass as Table I. Maximum likelihood parameter values from Salazar et al. [9] from Equations (1)-(4). Parameter Description Value rh,p Hal...

work page 2025

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J. Einasto, On the Construction of a Composite Model for the Galaxy and on the Determination of the System of Galactic Parameters, Trudy Astrofizicheskogo Instituta Alma-Ata 5, 87 (1965)

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This reduces the number of degrees of freedom in the individual fits to two, the same number of degrees of freedom in the Salazar et al

Diemer relies on the parametric model of Diemer [11], which takes the form ρorb(r) = ρs exp(S(r)) (23) where S(r) ≡ −2 α r rs α − 1 − 1 β " r rt β − rs rt β# .(24) We note that while the Diemer model has four degrees of freedom ( rt, rs, α, and β), when fitting individual halo profiles Diemer holds the parameters α and β fixed to their best-fit values in ...

work page

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S. More, B. Diemer, and A. V. Kravtsov, The Splashback- jacob.d.santiago2.mil@us.navy.mil Radius as a Physical Halo Boundary and the Growth of Halo Mass, Astrophys. J. 810, 36 (2015), arXiv:1504.05591 [astro-ph.CO]

arXiv 2015

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Garc ´ ıa, E

R. Garc ´ ıa, E. Salazar, E. Rozo, S. Adhikari, H. Aung, B. Diemer, D. Nagai, and B. Wolfe, A better way to define dark matter haloes, MNRAS 521, 2464 (2023), arXiv:2207.11827 [astro-ph.CO]

arXiv 2023

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J. F. Navarro, C. S. Frenk, and S. D. M. White, A Univer- sal Density Profile from Hierarchical Clustering, Astro- phys. J. 490, 493 (1997), arXiv:astro-ph/9611107 [astro- ph]

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A natural boundary of dark matter haloes revealed around the minimum bias and maximum infall locations

M. Fong and J. Han, A natural boundary of dark matter haloes revealed around the minimum bias and maximum infall locations, MNRAS 503, 4250 (2021), arXiv:2008.03477 [astro-ph.CO]

work page Pith review arXiv 2021