Dark Matter in Draco and Bo\"otes I: Hints of a Core in an Ultra-Faint Dwarf from Simulation-Based Inference

Andr\'es Ba\~nares-Hern\'andez; Andrew B. Pace; Claude-Andr\'e Faucher-Gigu\`ere; Hao Yang; Justin Read; Kevin McKinnon; Kohei Hayashi; Lina Necib; Nathan R. Sandford; Ting S. Li

REVIEW 2 major objections 2 minor 1 cited by

Dark Matter in Draco and Bo\"otes I: Hints of a Core in an Ultra-Faint Dwarf from Simulation-Based Inference

T0 review · 2 major / 2 minor · reviewed 2026-06-26 · grok-4.3

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Pith's one-line read GraphNPE recovers a low central density in Boötes I consistent with a dark matter core by using higher-order velocity moments.

desk verdict GraphNPE reduces bias in dwarf density recovery by using kurtosis and selection functions, but the low central density for Boötes I rests on an untested forward model match to the S5 survey. read the letter →

arxiv 2606.26218 v1 pith:Q26SLOPW submitted 2026-06-24 astro-ph.GA astro-ph.HE

Tri Nguyen , Lina Necib , Ting S. Li , Justin Read , Andrés Bañares-Hernández , Claude-André Faucher-Giguère , Kohei Hayashi , Kevin McKinnon

show 3 more authors

Andrew B. Pace Nathan R. Sandford Hao Yang

This is my paper · ORCID

classification astro-ph.GAastro-ph.HE

keywords darkmatterdensityprofilesdwarfspheroidalgalaxiessimulation-basedinferencevelocitykurtosisBoötesIDracodynamicalmodelingultra-faintdwarfs

topics Dark Matter

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 develops GraphNPE, a simulation-based inference method that forward-models stellar kinematics while including measurement uncertainties and spectroscopic selection functions. Standard approaches that use only line-of-sight velocity dispersion are shown on mocks to bias results toward cuspy dark matter profiles. Including line-of-sight kurtosis breaks key degeneracies and recovers the input profile with substantially less bias. When applied to real data, Draco yields a cuspy inner profile consistent with literature, while Boötes I yields a central density significantly lower than prior estimates and consistent with a core.

What carries the argument

GraphNPE, a simulation-based inference method that incorporates spectroscopic selection functions and measurement uncertainties in the forward model and uses line-of-sight kurtosis together with dispersion to constrain the dark matter density profile.

What would settle it

New observations of Boötes I with an independently verified selection function that, when fed through GraphNPE, produce a kurtosis consistent with a cuspy profile would falsify the core inference.

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Extended reading notes

Core claim

GraphNPE embeds the full survey selection function and error model inside a simulation-based forward model. Mock tests demonstrate that dispersion-only methods remain biased toward cusps even without mass-anisotropy degeneracy, whereas access to kurtosis allows recovery of the true density profile. Applied to Draco with two independent datasets, GraphNPE finds rho_150 approximately 1.6-1.9 times 10^8 solar masses per cubic kiloparsec and a marginally cuspy inner slope. For Boötes I it recovers rho_150 equal to 0.36 plus 0.15 minus 0.11 times 10^8 solar masses per cubic kiloparsec, lower than literature values and consistent with a cored profile.

Load-bearing premise

The forward model in GraphNPE correctly captures the full spectroscopic selection function and measurement uncertainties of the S5 survey for Boötes I and the MMT/DESI data for Draco, so that mismatches in higher-order moments arise from the dark matter profile rather than unmodeled observational effects.

Editorial extensions

If this is right

Dispersion-only Jeans modeling can fit the observed dispersion yet fail to reproduce the observed kurtosis.
Boötes I has one of the lowest central dark matter densities among dwarfs at comparable stellar mass.
J- and D-factors for indirect detection can be derived from the inferred profiles for both galaxies.
Higher-order velocity moments become essential once statistical power is limited.

Reading between the lines

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

Extending GraphNPE to additional ultra-faint dwarfs would test whether cored profiles are common at low stellar masses.
If the core signal survives improved selection modeling, it would increase tension with pure cold dark matter predictions for the smallest galaxies.
The method could be used to re-examine other dwarfs where dispersion-only analyses have produced unexpectedly high central densities.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The main takeaway is that GraphNPE recovers less biased inner densities than standard Jeans modeling by folding in kurtosis and the full selection function, and it produces a lower central density for Boötes I than previous work. The method itself is the clearest advance: mock tests show that dispersion-only fits pull toward cusps even without the usual mass-anisotropy problem, and adding the fourth moment helps. Application to Draco gives consistent results across two datasets and aligns with the literature cusp preference. For Boötes I the posterior sits at ρ_150 = 0.36^{+0.15}_{-0.11} × 10^8 M_⊙ kpc^{-3}, which would be notable if it holds.

The soft spot is exactly the one flagged in the stress test. The Boötes I claim depends on the forward model reproducing the S5 magnitude limits, fiber allocation, velocity errors, and completeness at the level needed to trust the kurtosis signal. The paper validates the approach on generic mocks, but nothing supplied confirms those mocks replicate the precise S5 characteristics for this ultra-faint system. With the abstract already noting limited statistical power, even modest mismatches could shift the inner slope. Draco looks more robust because the two independent datasets agree.

This paper is for people working on simulation-based inference for stellar kinematics or on the smallest dwarfs. The method is worth trying on other systems if the selection-function modeling can be shown to be accurate. It deserves peer review because the technical step is substantive and the Draco results are internally consistent; the Boötes I interpretation will need the referee to check the forward-model fidelity directly.

Referee Report

2 major / 2 minor

Summary. The paper introduces GraphNPE, a simulation-based inference method for dynamical mass modeling of dwarf spheroidals that incorporates measurement uncertainties and spectroscopic selection functions in the forward model. Mock tests show that including line-of-sight kurtosis reduces bias toward cuspy profiles relative to dispersion-only approaches. Applied to Draco (MMT/Hectochelle and DESI), results are consistent across datasets and marginally favor cuspy inner profiles; for Boötes I (S5 survey), the method recovers ρ_150 = 0.36^{+0.15}_{-0.11} × 10^8 M_⊙ kpc^{-3}, lower than literature values and consistent with a core, though statistical power is limited.

Significance. If the forward model is accurate, GraphNPE offers a route to less-biased density profiles in ultra-faint dwarfs by leveraging higher-order moments. The mock-data tests explicitly demonstrate reduced bias when kurtosis is included, and the multi-survey consistency for Draco provides a useful internal check. The Boötes I result, if robust, would place it among the lowest-density dwarfs at its stellar mass, with implications for dark-matter models. These elements are strengths of the work.

major comments (2)

[Application to Boötes I] The headline Boötes I result (ρ_150 = 0.36^{+0.15}_{-0.11} × 10^8 M_⊙ kpc^{-3}, consistent with a core) is load-bearing for the paper's claim of a low-density ultra-faint dwarf. This inference requires that the GraphNPE forward model exactly reproduces the S5 survey's magnitude limits, spatial sampling, fiber allocation, velocity error distribution, and position-dependent completeness; any mismatch would be misattributed to the dark-matter profile rather than observational effects. The manuscript states limited statistical power for Boötes I but does not provide explicit validation that the mocks replicate these precise S5 characteristics (as opposed to generic selection functions).
[Mock data tests] The mock-data tests demonstrate reduced bias when kurtosis is included, but the description indicates these are generic mocks. It is unclear whether the validation mocks incorporate the specific selection functions and uncertainty distributions of the S5 survey (or MMT/DESI) used for the real Boötes I and Draco data; this affects how directly the mock results support the real-data inferences.

minor comments (2)

[Draco DESI analysis] The abstract states that second-order Jeans modeling on DESI fits the dispersion but fails to reproduce the kurtosis; a quantitative measure of this failure (e.g., χ² or residual plot reference) would clarify the strength of the demonstration that higher-order moments are essential.
Notation for ρ_150 and the exact radial scale (150 pc) should be defined at first use in the main text for readers unfamiliar with the convention.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which help clarify the validation requirements for our forward-modeling approach. We address the major comments point by point below.

read point-by-point responses

Referee: [Application to Boötes I] The headline Boötes I result (ρ_150 = 0.36^{+0.15}_{-0.11} × 10^8 M_⊙ kpc^{-3}, consistent with a core) is load-bearing for the paper's claim of a low-density ultra-faint dwarf. This inference requires that the GraphNPE forward model exactly reproduces the S5 survey's magnitude limits, spatial sampling, fiber allocation, velocity error distribution, and position-dependent completeness; any mismatch would be misattributed to the dark-matter profile rather than observational effects. The manuscript states limited statistical power for Boötes I but does not provide explicit validation that the mocks replicate these precise S5 characteristics (as opposed to generic selection functions).

Authors: We agree that the manuscript would benefit from explicit validation showing that the forward model reproduces the S5 survey's specific characteristics. The current text describes incorporation of selection functions and uncertainties but does not include side-by-side quantitative comparisons (e.g., distributions of magnitudes, positions, velocities, or completeness maps) between mocks and S5 data. In the revised manuscript we will add an appendix with such comparisons to directly address this point. We already note the limited statistical power for Boötes I, which we will emphasize further in interpreting the low central density. revision: yes
Referee: [Mock data tests] The mock-data tests demonstrate reduced bias when kurtosis is included, but the description indicates these are generic mocks. It is unclear whether the validation mocks incorporate the specific selection functions and uncertainty distributions of the S5 survey (or MMT/DESI) used for the real Boötes I and Draco data; this affects how directly the mock results support the real-data inferences.

Authors: The mock tests use generic but realistic selection functions to isolate the effect of including higher-order moments on bias reduction, independent of any particular survey. This controlled demonstration supports the general validity of accessing kurtosis. For the real-data applications the GraphNPE forward model is customized to the exact selection functions and error distributions of each survey (MMT/Hectochelle, DESI, S5), as stated in the methods. We will revise the text to make this distinction explicit and to clarify that the generic mocks provide supporting evidence for the method rather than direct validation of the survey-specific inferences. revision: partial

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; GraphNPE is a validated SBI method with independent mocks

full rationale

The derivation uses simulation-based inference (GraphNPE) with a forward model that includes selection functions and uncertainties, validated on separate mock datasets before application to real Draco and Boötes I observations. No self-definitional steps, fitted inputs renamed as predictions, load-bearing self-citations, or ansatzes smuggled via citation appear in the abstract or described chain. The reported ρ_150 for Boötes I is a posterior inference from data, not a reduction to inputs by construction. The method remains self-contained against external benchmarks.

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

Abstract-only review; free parameters, axioms, and invented entities cannot be enumerated without the methods and modeling sections. The central result depends on the correctness of the GraphNPE forward model and the assumption that the mock galaxies adequately sample the relevant selection effects.

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Cite this review

Pith. "Pith review of Dark Matter in Draco and Bo\"otes I: Hints of a Core in an Ultra-Faint Dwarf from Simulation-Based Inference." pith.science (2026). https://pith.science/paper/Q26SLOPW

@misc{pith2026260626218,
  author       = {Pith},
  title        = {Pith review of: Dark Matter in Draco and Bo\"otes I: Hints of a Core in an Ultra-Faint Dwarf from Simulation-Based Inference},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q26SLOPW}},
  note         = {Machine review of arXiv:2606.26218}
}

abstract

The density profiles of dwarf spheroidal galaxies are among the most sensitive probes of dark matter physics, yet extracting them from noisy stellar kinematics remains a fundamental obstacle. We present GraphNPE, a simulation-based inference method for dynamical mass modeling that incorporates measurement uncertainties and spectroscopic selection functions in the forward model. Using mock data, we show that methods relying solely on line-of-sight velocity dispersion are biased toward cuspy density profiles, even in the absence of the mass-anisotropy degeneracy. By accessing higher-order velocity moments, particularly line-of-sight kurtosis, GraphNPE breaks key degeneracies and recovers density profiles with substantially less bias. We apply GraphNPE to Draco and Bo\"otes I using MMT/Hectochelle and DESI for Draco, and the S5 survey for Bo\"otes I. For each, we report density profiles and dark matter $J$- and $D$-factors. For Draco, GraphNPE yields consistent results across datasets, marginally preferring a cuspy inner profile ($\rho_{150} \sim 1.6-1.9 \times 10^8\,\mathrm{M}_\odot\,\mathrm{kpc}^{-3}$) in agreement with literature. On DESI, however, second-order Jeans modeling fits the dispersion but fails to reproduce the kurtosis, demonstrating higher-order moments are essential. For Bo\"otes I, limited statistical power prevents definitive determination of the inner slope. GraphNPE recovers $\rho_{150} = 0.36^{+0.15}_{-0.11} \times 10^8\,\mathrm{M}_\odot\,\mathrm{kpc}^{-3}$, significantly lower than literature and consistent with a cored inner profile. This places Bo\"otes I among the lowest density dwarfs at comparable stellar masses.

Figures

Figures reproduced from arXiv: 2606.26218 by the authors.

**Figure 1.** The LOSV (top), binned dispersion profiles (middle), and radial distributions of tracers (bottom) for Draco (left) and Boötes I (right). For Draco, the MMT (W23, blue circles) and DESI (D25, orange squares) datasets are shown. For Boötes I, the S 5 comb (S26, teal circles) and S 5 (S26, purple squares) datasets are shown. Gray vertical lines denote the half-light radius R1/2. In the middle panels, data points show t… view at source ↗

**Figure 2.** The LOSV measurement uncertainties ∆los as a function of projected radius for Draco (top) and Boötes I (bottom), with normalized distributions shown on the right. The tracers of each dataset are divided into multiple bins, with the median and 68% percentile of ∆los in each bin shown as data points with error bars. For Draco, the MMT (W23, blue circles) and DESI (D25, orange squares) datasets are shown. For Boötes I… view at source ↗

**Figure 3.** The projected radial distributions of spectroscopic tracers for Draco (top panel) and Boötes I (bottom panel). Filled histograms show the observed spectroscopic samples: MMT (W23) (blue) and DESI (D25) (orange) for Draco, and S 5 comb (teal) and S 5 (purple) S26 for Boötes I. Dashed histograms show the corresponding mock datasets after applying the empirical selection functions via resampling. The black solid cur… view at source ↗

Figures from the paper (10 more)

**Figure 4.** Figure 4: Comparison between the inferred profiles on mock galaxies with CoreOM parameters by GraphNPE and second-order Jeans modeling. From top to bottom, the rows show the DM density ρ(r), velocity anisotropy β(r), LOSV dispersion σlos(R), and LOSV kurtosis profiles κlos(R). F…

**Figure 5.** Figure 5: Same as [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]

**Figure 6.** Figure 6: The expected scatter in the intrinsic velocity dispersion estimator SD[ˆσintr], as a function of the number of tracer stars N and measurement uncertainties ∆, estimated empirically over M = 2000 Monte Carlo realizations. Two levels of intrinsic dispersion are shown, σi…

**Figure 7.** Figure 7: The LOSV dispersion σlos(R) (top) and kurtosis profiles κlos(R) (bottom) as a function of projected radius for varying inner slopes γ, computed for an isotropic velocity profile (β = 0). The DM halo parameters are chosen to match the Wolf mass of Draco (Wolf et al. 20…

**Figure 8.** Figure 8: From top to bottom: the inferred DM density ρ(r), velocity anisotropy β(r), LOSV dispersion σlos(R), and LOSV kurtosis κlos(R) profiles for Draco from GraphNPE and second-order Jeans modeling. The left and right columns show results for MMT (W23) and DESI (D25), respec…

**Figure 9.** Figure 9: Same as [PITH_FULL_IMAGE:figures/full_fig_p019_9.png]

**Figure 10.** Figure 10: The DM density at 150 pc, ρ150, for Draco (left) and Boötes I (right). Colored data points show measurements from this work, with filled and open symbols corresponding to the γ ∈ [−1, 2] and γ ∈ [0, 2] priors, respectively. Blue and pink points show results from Graph…

**Figure 11.** Figure 11: The J-factors (top row) and D-factors (bottom row) for Draco (left column) and Boötes I (right column). Colored data points show measurements from this work, with filled and open symbols corresponding to the γ ∈ [−1, 2] and γ ∈ [0, 2] priors, respectively. Blue and pi…

**Figure 12.** Figure 12: The inner DM density at 150 pc as a function of pre-infall halo mass M200, estimated by abundance matching (AM). For better visualization, we apply a small offset to Mam 200 of the same galaxy. Literature values are compiled from the following sources: Milky Way dwarf…

**Figure 13.** Figure 13: Inner slope at 1.5% of the virial radius rvir as a function of the stellar mass fraction M⋆/M200, with the pre-infall halo mass estimated by AM. For better visualization, we apply a small offset to M⋆/M200 of the same galaxy. The blue and orange bands show theoretical…

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Forward citations

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Pith tools

Reviewed June 26, 2026 · model on record in the stance chip above.

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