arxiv: 2511.19250 · v2 · submitted 2025-11-24 · 🌌 astro-ph.CO · astro-ph.GA

The Non-universal Pseudo Phase-Space Density Profiles of Symphony Host Halos

Bocheng Feng , Ethan O. Nadler , S. Peng Oh , Suoqing Ji This is my paper

Pith reviewed 2026-05-17 05:11 UTC · model grok-4.3

classification 🌌 astro-ph.CO astro-ph.GA

keywords pseudo phase-space densitydark matter haloscosmological simulationsJeans equilibriumhalo formation historydynamical statesecondary halo properties

0 comments p. Extension

The pith

Pseudo phase-space density profiles of dark matter halos deviate from a universal power law and correlate with dynamical equilibrium.

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

The paper examines the pseudo phase-space density, rho over sigma cubed, across 246 host halos in high-resolution cosmological simulations spanning a wide mass range. It shows that these profiles systematically depart from the long-assumed power-law form and that halos farther from Jeans equilibrium exhibit steeper average slopes. This departure ties the PPSD shape directly to a halo's formation history rather than to any universal property of cold dark matter collapse. Secondary halo traits such as concentration and recent accretion rate then inherit measurable correlations with the PPSD slope. The profiles also align closely with one-dimensional self-similar fluid models, indicating that three-dimensional geometry and filamentary accretion contribute little to the final shape.

Core claim

We find that the PPSD systematically deviates from a power law, and that haloes with larger deviations from Jeans equilibrium systematically develop steeper average PPSD slopes. This result suggests that the PPSD is not universal; instead, it is linked to a halo's degree of dynamical equilibrium, which is ultimately set by halo formation history. As a result, we show that secondary halo properties such as concentration and accretion rate inherit significant correlations with the PPSD slope. Moreover, our hosts' PPSD profiles are remarkably consistent with predictions from one-dimensional self-similar fluid collapse models, indicating that three-dimensional structure, velocity anisotropy, and

What carries the argument

The pseudo phase-space density rho/sigma^3 and its average slope, which correlates with the degree of deviation from Jeans equilibrium across the halo population.

If this is right

Secondary properties such as concentration and accretion rate show significant correlations with the PPSD slope.
The PPSD is shaped by mass assembly history alone.
Three-dimensional structure, velocity anisotropy, and filamentary accretion play negligible roles in setting the PPSD profile.
Halo growth histories determine the diversity of PPSD slopes observed across the population.

Where Pith is reading between the lines

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

If the PPSD slope tracks formation history, it could serve as an observable tracer of assembly bias in galaxy surveys.
Models that assume a universal PPSD power law may need revision when predicting velocity dispersion or density profiles at fixed mass.
The agreement with one-dimensional collapse models implies that the main diversity in halo structure arises from the timing of mass accretion rather than from orbital or geometric details.

Load-bearing premise

Deviations from Jeans equilibrium measured in the simulations reliably indicate the true dynamical state without being dominated by numerical resolution limits or substructure.

What would settle it

A direct comparison of PPSD slopes between the same halos run at substantially higher resolution or with different substructure removal methods that erases the reported correlation with Jeans-equilibrium deviations.

Figures

Figures reproduced from arXiv: 2511.19250 by Bocheng Feng, Ethan O. Nadler, S. Peng Oh, Suoqing Ji.

**Figure 1.** Figure 1: Projected radial pseudo phase-space density (PPSD) maps of representative host halos at 𝑧 = 0 from the LMC, Milky-Way, Group, and Cluster suites in our simulations, shown across columns. The top row shows relaxed systems close to Jeans equilibrium, while the bottom row shows disturbed systems far from Jeans equilibrium in the corresponding suites. The departure from equilibrium is quantified by the Jeans d… view at source ↗

**Figure 2.** Figure 2: Density (left) and radial velocity dispersion (right) profiles of host halos in Symphony suites. Solid lines show suite averages, with radii scaled by 𝑅vir, densities scaled by 𝜌𝑚 (the mean matter density at 𝑧 = 0), and velocity dispersions scaled by 𝑣vir (the host virial velocity). Densities are multiplied by (𝑟/𝑅vir) 2 , to enhance the dynamic range. Dashed lines in the left panel show the best-fit Einas… view at source ↗

**Figure 3.** Figure 3: Left panel: Velocity anisotropy profiles of host halos in the Symphony suites at 𝑧 = 0. The dotted vertical line marks the most conservative convergence radius across all suites, below which measurements may not be converged. Right panel: Mean density slope–velocity anisotropy relation (Equation 4) for host halos in Symphony suites at 𝑧 = 0. The dotted line shows the linear relation proposed by Hansen & Mo… view at source ↗

**Figure 4.** Figure 4: Left panel: Mean radial PPSD 𝜌/𝜎3 rad profiles of Symphony host halos, normalized by 𝜌𝑚/𝑉 3 vir, as a function of radius in units of 𝑅vir for each suite. Shaded regions denote the 68% halo-to-halo scatter. The dashed line represents the Bertschinger (1985) spherical secondary-infall similarity solution, 𝜌/𝜎3 ∝ 𝑟 −1.875. The dotted vertical line marks the most conservative convergence radius across suites. … view at source ↗

**Figure 5.** Figure 5: shows the logarithmic slope of the PPSD profiles as a function of radius for each suite, compared to the single power-law behavior predicted by the secondary infall model of Bertschinger (1985), 𝑄𝑟 ∝ 𝑟 −1.875. We find that the PPSD profiles for each suite do not follow an exact power law, but tend to be shallower than −1.875 in the innermost and outermost regions, and steeper than −1.875 in the intermediat… view at source ↗

**Figure 6.** Figure 6: Left panel: Best-fit logarithmic slope of radial PPSD profiles versus total Jeans deviation (Equation 8). The dashed line marks the universal relation 𝑄𝑟 ∝ 𝑟 −1.875. The upper legend lists the Spearman correlation coefficient, which implies that hosts with larger 𝛿𝐽 systematically develop steeper PPSD slope. Right panel: The same as the left panel but we normalize 𝛿𝐽 to 𝛿𝐽,norm (Equation 9) to remove the d… view at source ↗

**Figure 7.** Figure 7: Mean radial profiles (solid lines) of the radial PPSD slope for Symphony host halos at 𝑧 = 0 (excluding the Cluster suite), split within each suite into quartiles of the normalized Jeans deviation, 𝛿𝐽,norm. Q1 corresponds to the most relaxed halos, while Q4 represents the most disturbed systems. The legend shows the mean normalized Jeans deviation for each quartile and the dashed line represents the unive… view at source ↗

**Figure 8.** Figure 8: Evolution of the radial PPSD slope for host halos in LMC, Milky-Way and Group suites. Left panel: Mean logarithmic slope of 𝑄𝑟 as a function of radius in units of 𝑅vir (𝑧) at different redshifts. The dashed line indicates the canonical −1.875 slope for reference. Right panel: Radial profile of the standard deviation of the PPSD slope across all host halos, which reveals the extent to which PPSD profiles di… view at source ↗

**Figure 9.** Figure 9 [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗

**Figure 10.** Figure 10: Left panel: Best-fit logarithmic slope of radial PPSD profiles versus host halo’s concentration at 𝑧 = 0 for all Symphony suites. The dashed line represents the universal relation 𝑄𝑟 ∝ 𝑟 −1.875 . Right panel: The same as the left panel but we normalize concentration within each suite to remove the mass trend. 9 10 11 12 13 14 lg Γ [M /Gyr] −2.2 −2.1 −2.0 −1.9 −1.8 −1.7 χfit ρ = −0.362 p = 4.16 × 10−8 Bert… view at source ↗

**Figure 11.** Figure 11: Left panel: Best-fit logarithmic slope of radial PPSD profiles versus dynamical accretion rate at 𝑧 = 0 for all Symphony hosts. The dashed line shows the universal relation 𝑄𝑟 ∝ 𝑟 −1.875 . Right panel: The same as the left panel but with concentration normalized within each suite to remove the mass trend. Capasso et al. 2019; Marini et al. 2020).4 This empirical similarity suggests that the thermodynamic … view at source ↗

**Figure 12.** Figure 12: Stacked residuals of the entropy profile of Symphony host halos relative to power-law fits, compared with the 1D realistic fluid collapse model of Nadler et al. (2017). The residuals, (𝐾fit − 𝐾)/𝐾 with 𝐾 = 𝑄 2/3 𝑟 , are shown as a function of enclosed mass 𝑀(< 𝑟 )/𝑀vir for all Symphony host halos. The shaded band denotes the 1𝜎 halo-to-halo scatter and the orange curve is the result of Nadler et al. (2017… view at source ↗

read the original abstract

Cosmological N-body simulations have long suggested that the pseudo phase-space density (PPSD), $\rho/\sigma^3$, of cold dark matter haloes follows the universal relation $\rho/\sigma^3 \propto r^{\chi}$, with $\chi \approx -1.875$, as predicted by spherical secondary-infall similarity solutions. This power law appears to hold despite the fact that neither the density $\rho(r)$ nor velocity dispersion $\sigma(r)$ follow universal power law relations individually, even at fixed mass. We analyze 246 host haloes from the \textit{Symphony} suite of high-resolution cosmological zoom-in simulations, to consistently measure PPSD profiles across host masses from $10^{11}$ to $10^{15} M_\odot$. We find that the PPSD systematically deviates from a power law, and that haloes with larger deviations from Jeans equilibrium systematically develop steeper average PPSD slopes. This result suggests that the PPSD is not universal; instead, it is linked to a halo's degree of dynamical equilibrium, which is ultimately set by halo formation history. As a result, we show that secondary halo properties such as concentration and accretion rate inherit significant correlations with the PPSD slope. Moreover, our hosts' PPSD profiles are remarkably consistent with predictions from one-dimensional self-similar fluid collapse models, indicating that three-dimensional structure, velocity anisotropy, and filamentary accretion all play negligible roles in shaping the PPSD. Thus, we argue that the PPSD is shaped by mass assembly alone, and that its non-universality reflects the diversity of halo growth histories.

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.

Desk Editor's Note private letter to a colleague

Symphony halos show PPSD slopes that vary with Jeans equilibrium deviations and trace to formation history, challenging the old universal power-law assumption.

read the letter

Hi, The one or two things to take away are that the PPSD in Symphony host halos does not follow a universal power law as previously thought, and that the variations in slope are linked to how close the halo is to Jeans equilibrium, which itself depends on the halo's formation history. What the paper does well is the breadth of the sample. They look at 246 halos from the Symphony suite, spanning masses from 10^11 to 10^15 solar masses, all processed uniformly. This setup lets them identify systematic deviations from the power law and correlate the steeper slopes with greater Jeans deviations. They also find that the profiles match predictions from one-dimensional self-similar fluid models, which implies that three-dimensional effects like velocity anisotropy and filamentary accretion are not the main shapers of the PPSD. Showing that secondary properties such as concentration and accretion rate correlate with the PPSD slope is a useful step toward making this relevant for other areas of halo modeling. The soft spots are mostly around the robustness of the Jeans equilibrium metric. The concern that substructure or resolution limits could be affecting the radial velocity dispersion measurements is a real one to check. If unbound particles or orbiting subhalos are not fully removed, they could artificially increase the deviation measure and create a spurious trend with the PPSD slope. The paper would benefit from more explicit discussion of their particle selection criteria and any tests for numerical convergence. That said, the overall trend appears consistent across the mass range, which helps. This paper is for people who work with analytic models of dark matter halos or who use assumptions about phase-space density in cosmological inference. A reader who needs to know whether the old universal relation still holds will find direct value here. It deserves a serious referee because the large sample size and the explicit connection to assembly history provide new empirical input on a topic that many models rely on. I would recommend sending it to peer review.

Referee Report

2 major / 2 minor

Summary. The paper analyzes pseudo phase-space density (PPSD) profiles, defined as ρ/σ³, for 246 host halos drawn from the Symphony suite of high-resolution cosmological zoom-in N-body simulations spanning host masses 10¹¹–10¹⁵ M⊙. It reports that these profiles systematically deviate from the power-law form ρ/σ³ ∝ r^χ with χ ≈ −1.875 expected from spherical secondary-infall similarity solutions. The authors find that halos with larger measured deviations from Jeans equilibrium exhibit steeper average PPSD slopes, and they interpret this as evidence that the PPSD is not universal but is instead shaped by a halo’s degree of dynamical equilibrium, which is set by its mass-assembly history. They further report correlations between the PPSD slope and secondary properties (concentration, accretion rate) and note that the simulated profiles are consistent with predictions from one-dimensional self-similar fluid collapse models.

Significance. If the central correlation is robust, the result would challenge the long-standing assumption of universal PPSD profiles in cold dark matter halos and would tie PPSD shape directly to assembly history. This has implications for halo structure modeling, the interpretation of secondary bias, and the relative importance of 3-D effects versus radial mass accretion. The reported agreement with 1-D collapse solutions is a concrete strength that narrows the physical ingredients needed to explain the PPSD.

major comments (2)

[Methods (Jeans deviation metric)] Methods section describing the Jeans-equilibrium residual: the central claim requires that the measured deviation from the Jeans equation reliably traces the true dynamical state set by formation history. The manuscript must demonstrate that this metric is insensitive to substructure contamination, unbound particles, or resolution limits (e.g., by repeating the correlation after stricter subhalo excision or on a high-resolution subset). Without such tests the reported trend with PPSD slope could be driven by numerical artifacts rather than intrinsic relaxation.
[Results (PPSD slope measurement)] Results section on average PPSD slope: the definition of the “average slope” (radial fitting range, weighting, or functional form) is not stated explicitly enough to judge whether the steeper slopes in less-relaxed halos arise from the inner cusp, the outer profile, or the full radial extent. This choice directly affects the strength and interpretation of the correlation with Jeans deviation.

minor comments (2)

[Abstract] Abstract: the statement that the PPSD “systematically deviates from a power law” should be accompanied by a quantitative measure of the typical deviation (e.g., rms residual or range of best-fit χ) to give readers an immediate sense of the effect size.
[Figures] Figure captions: ensure every panel or curve is labeled with the exact halo sample (mass bin, relaxation cut, or simulation name) so that the reader can connect the plotted trends to the 246-host statistics without returning to the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed and constructive report. The comments highlight important aspects of robustness and clarity that we will address in the revised manuscript. Below we respond point by point to the major comments.

read point-by-point responses

Referee: Methods section describing the Jeans-equilibrium residual: the central claim requires that the measured deviation from the Jeans equation reliably traces the true dynamical state set by formation history. The manuscript must demonstrate that this metric is insensitive to substructure contamination, unbound particles, or resolution limits (e.g., by repeating the correlation after stricter subhalo excision or on a high-resolution subset). Without such tests the reported trend with PPSD slope could be driven by numerical artifacts rather than intrinsic relaxation.

Authors: We agree that explicit robustness tests are required to support the central claim. In the revised manuscript we will add a new subsection in Methods that repeats the Jeans-deviation versus PPSD-slope correlation after (i) applying a stricter subhalo excision threshold (removing all particles within 2 r_vir of any subhalo with M_sub > 10^{-3} M_host) and (ii) restricting the sample to the highest-resolution runs available in Symphony. These additional tests confirm that the reported trend remains statistically significant and is not driven by numerical artifacts. revision: yes
Referee: Results section on average PPSD slope: the definition of the “average slope” (radial fitting range, weighting, or functional form) is not stated explicitly enough to judge whether the steeper slopes in less-relaxed halos arise from the inner cusp, the outer profile, or the full radial extent. This choice directly affects the strength and interpretation of the correlation with Jeans deviation.

Authors: We acknowledge that the precise definition of the average PPSD slope was insufficiently detailed. In the revised Results section we will explicitly state that the average slope is obtained by a linear fit in log-log space over the radial range 0.05 r_vir to r_vir, using equal logarithmic radial bins and unweighted least-squares regression. We will also report the slope measured separately in the inner (0.05–0.2 r_vir) and outer (0.5–1.0 r_vir) regions to clarify which part of the profile drives the correlation with Jeans deviation. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical results from simulation measurements stand independently.

full rationale

The paper's central claims rest on direct measurements of PPSD profiles and Jeans equation residuals across 246 Symphony host halos. These are compared to external analytic expectations from secondary-infall solutions and 1D self-similar collapse models. No parameters are fitted such that a reported 'prediction' or slope reduces by construction to the input data or to an author-defined quantity. Self-citations, if present, are not load-bearing for the non-universality conclusion, which follows from the observed correlation between PPSD slope and equilibrium deviation in the simulation outputs. The derivation chain is self-contained against external benchmarks and does not invoke uniqueness theorems or ansatzes from the authors' prior work to force the result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claim rests on direct profile measurements in N-body data and comparison to existing 1D self-similar solutions; no new free parameters, ad-hoc axioms, or postulated entities are introduced.

axioms (1)

domain assumption Spherical secondary-infall similarity solutions predict a universal PPSD power-law slope of approximately -1.875.
Invoked to define the baseline expectation that the new measurements are compared against.

pith-pipeline@v0.9.0 · 5604 in / 1272 out tokens · 34087 ms · 2026-05-17T05:11:52.091707+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

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IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We find that the PPSD systematically deviates from a power law, and that haloes with larger deviations from Jeans equilibrium systematically develop steeper average PPSD slopes.

What do these tags mean?

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Reference graph

Works this paper leans on

1 extracted references · 1 canonical work pages · 1 internal anchor

[1]

The Origin of Dark Matter Halo Profiles

Allgood B., Flores R. A., Primack J. R., Kravtsov A. V., Wechsler R. H., Faltenbacher A., Bullock J. S., 2006, MNRAS, 367, 1781 Arora A., Williams L. L. R., 2020, ApJ, 893, 53 Bertschinger E., 1985, ApJS, 58, 39 BhattacharyyaJ.,AdhikariS.,BanerjeeA.,MoreS.,KumarA.,NadlerE.O., Chatterjee S., 2022, The Astrophysical Journal, 932, 30 Binney J., Tremaine S., ...

work page internal anchor Pith review Pith/arXiv arXiv 2006