Pith. sign in

REVIEW 2 major objections 4 minor 67 references

JFlow: Model-Independent Spherical Jeans Analysis using Equivariant Continuous Normalizing Flows

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read JFlow replaces parametric stellar and dark-matter profiles with an equivariant continuous normalizing flow, recovering spherical Jeans-equation mass profiles from projected positions and line-of-sight velocities given an anisotropy input.

desk verdict JFlow is a clean, honest proof-of-concept for free-form spherical Jeans analysis, but its 'model-independent' claim needs an explicit caveat about the Gaussian velocity assumption and a non-Gaussian stress test. read the letter →

arxiv 2505.00763 v2 pith:GC3SLF64 submitted 2025-05-01 astro-ph.GA astro-ph.COcs.LGhep-exhep-ph

Sung Hak Lim , Kohei Hayashi , Shun'ichi Horigome , Shigeki Matsumoto , Mihoko M. Nojiri This is my paper · ORCID
classification astro-ph.GAastro-ph.COcs.LGhep-exhep-ph
keywords sphericalJeansequationdwarfspheroidalgalaxiesdarkmatterdensitynormalizingflowsequivariantvelocityanisotropymass-anisotropydegeneracyunsupervisedmachinelearning
topics Dark Matter
open problems 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

JFlow is an unsupervised machine-learning method for solving the spherical Jeans equation—the equilibrium condition linking stellar motions to enclosed mass—without assuming analytic forms for the stellar density, velocity dispersion, or dark-matter halo. From projected stellar positions and line-of-sight velocities, it trains an equivariant continuous normalizing flow to learn the spherically symmetric stellar number density $n(r)$ and a conditional flow to learn the radial velocity dispersion $\overline{v_r^2}(r)$. Inserting these into the Jeans equation yields the enclosed dark-matter mass $M(r)$, provided the velocity-anisotropy profile $\beta(r)$ is supplied as input. On a mock spherical dwarf-galaxy system with only 1,000 tracer stars, the method recovers $n(r)$, $\overline{v_r^2}$, $M(r)$, and the dark-matter density within about $2\sigma$ over the sampled radial range. A sympathetic reader would care because this replaces profile-choice bias in dwarf dark-matter inference, letting halo shapes beyond standard cusps and cores be tested directly.

What carries the argument

The load-bearing object is an equivariant continuous normalizing flow: a neural ODE $\frac{d\vec r_t}{dt} = \hat r \tanh(|\vec r_t|/r_s) F_r(|\vec r_t|,t;\theta)$ that evolves only the radial coordinate of an initially Gaussian sample, so the learned 3D stellar density is spherically symmetric by construction. A power-law warp $T_{\rm pow}: |\vec r|\to|\vec r|^{c+1}$ is composed with the cored flow to allow cuspy profiles, handling both flat-core and power-law stellar distributions. A companion conditional linear flow multiplies velocities by a diagonal matrix $L(r;\theta)$ whose diagonal entries are neural-network models of the radial and tangential velocity dispersions, with the tangential components slaved to the input $\beta(r)$. Training is done without an explicit projected likelihood: generated 3D samples are projected to observable coordinates, smoothed by a Gaussian kernel, and compared with kernel-smeared data through a Kullback-Leibler loss, which is differentiable so the flow parameters can be optimized. The spherical Jeans equation then is the bridge from these learned density and dispersion functions to $M(r)$ and $\rho_{\rm DM}(r)$.

What would settle it

Take a mock spherical system with a known cored or cuspy dark-matter profile and a deliberately strong tangential anisotropy, supply JFlow with the true $\beta(r)$, and check whether the recovered $\rho_{\rm DM}(r)$ matches the input over the well-sampled radial range; the paper already reports one $2.5\sigma$ deviation at $r\simeq0.4$ kpc, so repeating the test with larger tracer samples would reveal whether that deviation is finite-sample noise or a systematic bias that would invalidate the model-independent claim.

Watch

Extended reading notes

Core claim

The paper's central claim is that the three ingredients of spherical Jeans analysis—stellar number density, radial velocity dispersion, and the resulting enclosed mass—can be estimated from projected sky positions and line-of-sight velocities alone, with no parametric model for the stellar or dark-matter distributions. The flow is rotationally equivariant by construction: the base Gaussian is spherically symmetric and the learned ODE only moves points along the radial direction, so the modeled density is guaranteed spherical. A separate position-dependent linear flow parameterizes the velocity variance, and both models are trained by a likelihood-free loss that compares kernel-smoothed generated samples with kernel-smeared data in the projected observable space. With $\beta(r)$ given, the Jeans equation $M(r) = -\frac{r\overline{v_r^2}}{G}\left(\frac{d\ln(n\overline{v_r^2})}{d\ln r} + 2\beta\right)$ converts the learned functions into the dark-matter mass profile. On a mock cuspy-stellar, cored-dark-matter halo with 1,000 tracers, JFlow's estimates agree with the true profiles within about $2\sigma$, and the paper argues this carries over to other spherical systems and to testing nonstandard halo physics.

Load-bearing premise

The velocity-anisotropy profile $\beta(r)$ is provided as an external input and is never constrained by the data, so an incorrect $\beta$ propagates directly into biased dark-matter mass and density estimates except near the radius where the enclosed mass is nearly $\beta$-independent.

Editorial extensions

If this is right

  • Dark-matter density profiles of dispersion-supported dwarf galaxies can be inferred from a few hundred to a thousand member stars without committing to a halo model, improving the statistical reach of small spectroscopic samples.
  • Because no functional form is imposed, inferred halos can show cores, cusps, or other departures from standard profiles, making JFlow a direct test of self-interacting and wave dark-matter predictions.
  • Enclosed mass near the radius where the Jeans expression is nearly independent of $\beta$ remains reliable even when the input anisotropy is wrong, giving a robust anchor for mass estimates.
  • The likelihood-free KDE training strategy does not rely on analytic Abel projection, so the same machinery can be extended to non-spherical systems and to observing conditions where projection is complicated, as the paper anticipates.
  • Parametric fits that happen to include the true profile give narrower credible intervals, but JFlow avoids the bias that appears when the true profile lies outside the assumed family; with correct $\beta$, it estimates the mass profile without this bias.

Reading between the lines

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

  • A decisive practical extension would be to feed JFlow anisotropy profiles derived from independent data—proper motions, higher-order velocity moments, or multi-component modelling—because the paper itself shows that an incorrect $\beta$ biases the mass and density except near the special radius.
  • The reported $2.5\sigma$ deviation in mass density at $r\simeq0.4$ kpc is a candidate systematic tied to the KDE bandwidth and the curvature of $\overline{v_r^2}$ near the training-data boundary; a testable prediction is that increasing sample size or reducing bandwidth should shrink this deviation if it is statistical.
  • The same architecture could be used to estimate dark-matter density slopes for indirect-detection $J$-factors, where the inner density profile matters most and where free-form estimates would quantify the model uncertainty that parametric fits understate.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper introduces JFlow, an unsupervised machine-learning method for spherical Jeans analysis of dwarf spheroidal galaxies. It models the stellar number density n(r) with equivariant continuous normalizing flows and the radial velocity dispersion v2_r(r) with a conditional Gaussian flow, then substitutes the fitted functions into the spherical Jeans equation (Eq. 3) to estimate the enclosed mass M(r) and dark matter density ρ(r), given an input anisotropy profile β(r). Training uses only projected positions and line-of-sight velocities; the loss (Eq. 12) is the Kullback-Leibler divergence between kernel-smoothed projected densities of the data and the model. A proof-of-concept on the NonplumCoreIso Gaia Challenge mock with 1,000 tracers recovers the stellar density, radial velocity dispersion, enclosed mass, and mass density within 2σ over most of the radial range, and a comparison with parametric fitting shows wider credible intervals for JFlow, as expected for a free-form model.

Significance. If robust, JFlow would be a valuable free-form alternative to parametric Jeans analyses, enabling exploration of non-standard dark matter profiles without imposing analytic forms. The KDE-based training on projected data is a novel and asymptotically consistent approach, and the use of equivariant flows guarantees spherical symmetry by construction. The explicit treatment of the mass-anisotropy degeneracy via input β, and the demonstration that the enclosed-mass estimate near the Wolf radius is insensitive to β mismatch, are strengths. However, the 'model-independent' claim is overstated: the method also implicitly assumes a Gaussian velocity distribution at each radius, and it requires β as an input; the Gaussian assumption is not listed among the limitations and is not stress-tested in the paper.

major comments (2)
  1. [II (Eqs. 10–11) and III (Eq. 12)] The velocity model in Eqs. (10)–(11) is a position-dependent Gaussian with diagonal covariance, and the training loss (Eq. 12) is a full-distribution KL divergence between smoothed projected densities. In the asymptotic limit, JFlow therefore converges to the KL projection of the true phase-space density onto this Gaussian-velocity family. The spherical Jeans equation constrains only the second velocity moments, and there is no general guarantee that this KL projection preserves σ_r^2(r) when the true local velocity distribution is non-Gaussian. The paper validates the method only on the NonplumCoreIso mock and reports no test with deliberately non-Gaussian velocities; the Gaussian assumption is not listed among the limitations. Since the inferred M(r) and ρ(r) via Eq. (3) depend directly on σ_r^2(r), this is load-bearing for the central 'model-independent' claim. I request a non-Gaussian mock test (e.g., a two-component Gaussian or a distribution with non-zero excess kurtosis but identical σ_r^2(r) and β) and a quantification of the resulting bias; if bias appears, either relax the Gaussian velocity model or state the Gaussian assumption explicitly as a limitation in the abstract and conclusions.
  2. [III (near Eq. 12)] The statement that 'the optimal solution of the training in the asymptotic limit is f⊥∗Kh = f̂⊥∗Kh, and f̂⊥ will converge to f⊥ in the case of Gaussian kernels' is only true if the true projected density f⊥ is representable by the model family. Because the velocity model is Gaussian, this representability is not guaranteed for arbitrary dSph data; the general convergence target is the KL projection of f⊥ onto the family, not f⊥ itself. Please qualify this sentence accordingly, e.g., 'converges to the closest density within the model family.'
minor comments (4)
  1. [II, Eq. (2)] There is a typo: 'resplectively' should be 'respectively'.
  2. [II, Eq. (9)] The relation between the cusp parameter c and the inner logarithmic slope of the stellar density, namely d log n/d log r ≈ −3c/(c+1) at small r, is used implicitly in the iterative training of c in Appendix B but is never written out in the main text. Stating this explicitly would improve clarity.
  3. [III and Appendix A] Please state whether the NonplumCoreIso mock's intrinsic velocity distribution is Gaussian. This is directly relevant to the robustness test requested in Major Comment 1 and to interpreting the validation results.
  4. [Appendix B] The paper does not mention whether the code and trained models will be released. Given the complexity of the method, a code-availability statement would aid reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the enclosed mass is computed algebraically from independently fitted stellar density and velocity dispersion via the spherical Jeans equation.

full rationale

The central inference chain is self-contained rather than circular. JFlow trains neural density models to match the smoothed projected phase-space distribution through the KL-divergence loss in Eq. (12); the learned quantities are the stellar number density n(r) and radial velocity dispersion v2r(r). The enclosed mass is then obtained algebraically from the spherical Jeans equation, Eq. (3), M(r) = -r v2r/G [d ln(n v2r)/d ln r + 2β], with β supplied as an explicit external input. Neither the loss function nor the network architecture contains M(r), ρ(r), or the dark-matter profile as a fitting target, so the mass estimate is not a fitted parameter renamed as a prediction. The paper transparently acknowledges the mass-anisotropy degeneracy and states that β cannot be estimated from line-of-sight data alone, then tests sensitivity to an incorrect β, confirming that the method is conditional on β rather than circularly assuming it. The cusp parameter c in Eq. (9) is iteratively estimated from the model's own density slopes in Appendix B, making it data-driven rather than imposed from the true profile. Validation is performed against the external Gaia Challenge NonplumCoreIso mock, whose true stellar and dark-matter profiles are not used as training targets. The Gaussian conditional velocity model in Eqs. (10)-(11) and the assumptions of spherical symmetry and equilibrium are model assumptions that could affect accuracy on non-Gaussian or non-spherical systems, but they are not instances of a derivation reducing to its own inputs. Self-citations such as Refs. [17], [21], and [25] are contextual and not load-bearing for the central Jeans-inversion result, which rests on standard Abel inversion and the Jeans equation verified against external benchmarks.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central inference rests on standard Jeans kinematics, a Gaussian velocity likelihood, and the expressiveness of normalizing flows. No new physical entities are introduced. The free parameters, cusp exponent c, transition scale rs, and KDE bandwidth h, are either fitted to data or chosen by rule-of-thumb and directly affect the inferred density and dispersion profiles.

free parameters (3)
  • Cusp exponent c = Iteratively estimated (mean cusp exponent gamma_bar; c = gamma_bar/(3-gamma_bar))
    Controls the inner logarithmic slope of the stellar density via Eq. (9). Estimated from the data in a 20-step loop in Appendix B, not derived from first principles. It directly shapes n(r) and therefore affects the Jeans inversion.
  • Transition scale rs = N^{-1/5} for standardized data
    Suppresses the radial transformation below this scale in Eq. (8). Chosen by rule-of-thumb bandwidth, not fitted. It defines the region where the model behaves as a cored Gaussian and affects the stellar density estimate near the center.
  • KDE bandwidth h = N^{-1/(4+D)} for standardized data
    Smoothing scale for the projected likelihood and training data in Appendix B. Chosen by rule-of-thumb. The asymptotic consistency of the loss in Eq. (12) depends on h shrinking appropriately, and finite-sample bias varies with this choice.
assumptions (4)
  • domain assumption The stellar system is spherically symmetric and in steady-state dynamical equilibrium, so the spherical Jeans equation (Eq. 2) applies.
    Invoked in Section I to derive Eq. (2) and used throughout; if violated, M(r) from Eq. (3) is biased.
  • domain assumption The velocity distribution is Gaussian with position-dependent covariance L(r) given by Eq. (10).
    The conditional linear flow models a Gaussian; the projected likelihood is a KDE of samples from this Gaussian. If the true LOSVD is strongly non-Gaussian, the fitted second moments may be biased in finite samples.
  • standard math The base distribution is a 3D standard Gaussian, and the radial ODE family (Eq. 8), optionally composed with the power-law map (Eq. 9), can represent any relevant spherically symmetric stellar density.
    Relies on the expressiveness of continuous normalizing flows with a radial equivariant vector field; this is the standard foundation of normalizing flows (refs [15,38-42]).
  • standard math The KDE-smoothed likelihood converges to the true projected likelihood as sample size grows; the loss in Eq. (12) is the KL divergence between smoothed densities.
    Standard KDE consistency and KL minimization; the paper uses this to justify unbiased training (text around Eq. 12).

how reviews work

0 comments
Cite this review

Pith. "Pith review of JFlow: Model-Independent Spherical Jeans Analysis using Equivariant Continuous Normalizing Flows." pith.science (2026). https://pith.science/paper/GC3SLF64

@misc{pith2026250500763,
  author       = {Pith},
  title        = {Pith review of: JFlow: Model-Independent Spherical Jeans Analysis using Equivariant Continuous Normalizing Flows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GC3SLF64}},
  note         = {Machine review of arXiv:2505.00763}
}
read the original abstract

The kinematics of stars in dwarf spheroidal galaxies have been studied to understand the structure of dark matter halos. However, the kinematic information of these stars is often limited to celestial positions and line-of-sight velocities, making full phase space analysis challenging. Conventional methods rely on projected analytic phase space density models with several parameters and infer dark matter halo structures by solving the spherical Jeans equation. In this paper, we introduce an unsupervised machine learning method for solving the spherical Jeans equation in a model-independent way as a first step toward model-independent analysis of dwarf spheroidal galaxies. Using equivariant continuous normalizing flows, we demonstrate that spherically symmetric stellar phase space densities and velocity dispersions can be estimated without model assumptions. As a proof of concept, we apply our method to Gaia challenge datasets for spherical models and measure dark matter mass densities for given velocity anisotropy profiles. Our method can identify halo structures accurately, even with a small number of tracer stars.

Figures

Figures reproduced from arXiv: 2505.00763 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic diagram of different approaches for mod [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Maximum a posteriori estimation results for dSph parameters of [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

67 extracted references · 33 canonical work pages

  1. [1]

    M. W. Goodman and E. Witten, Detectability of Certain Dark Matter Candidates, Phys. Rev. D 31, 3059 (1985)

    work page 1985

  2. [2]

    Marrod´ an Undagoitia and L

    T. Marrod´ an Undagoitia and L. Rauch, Dark matter direct-detection experiments, J. Phys. G 43, 013001 (2016), arXiv:1509.08767 [physics.ins-det]

    arXiv 2016

  3. [3]

    Schumann, Direct Detection of WIMP Dark Mat- ter: Concepts and Status, J

    M. Schumann, Direct Detection of WIMP Dark Mat- ter: Concepts and Status, J. Phys. G 46, 103003 (2019), arXiv:1903.03026 [astro-ph.CO]

    arXiv 2019

  4. [4]

    Lin, Dark matter models and direct detection, PoS 333, 009 (2019), arXiv:1904.07915 [hep-ph]

    T. Lin, Dark matter models and direct detection, PoS 333, 009 (2019), arXiv:1904.07915 [hep-ph]

    arXiv 2019

  5. [5]

    T. R. Slatyer, Indirect detection of dark matter., in The- oretical Advanced Study Institute in Elementary Particle Physics: Anticipating the Next Discoveries in Particle Physics (2018) pp. 297–353, arXiv:1710.05137 [hep-ph]

    arXiv 2018

  6. [6]

    Hooper, TASI Lectures on Indirect Searches For Dark Matter, PoS T ASI2018, 010 (2019), arXiv:1812.02029 [hep-ph]

    D. Hooper, TASI Lectures on Indirect Searches For Dark Matter, PoS T ASI2018, 010 (2019), arXiv:1812.02029 [hep-ph]

    arXiv 2019

  7. [7]

    J. F. Navarro, C. S. Frenk, and S. D. M. White, A Uni- versal density profile from hierarchical clustering, Astro- phys. J. 490, 493 (1997), arXiv:astro-ph/9611107

    arXiv 1997

  8. [8]

    J. F. Navarro, C. S. Frenk, and S. D. M. White, Simula- tions of x-ray clusters, Mon. Not. Roy. Astron. Soc. 275, 720 (1995), arXiv:astro-ph/9408069

    arXiv 1995

Show all 67 references
  1. [9]

    J. F. Navarro, C. S. Frenk, and S. D. M. White, The Structure of cold dark matter halos, Astrophys. J. 462, 563 (1996), arXiv:astro-ph/9508025

    1996 arXiv

  2. [10]

    Tulin and H.-B

    S. Tulin and H.-B. Yu, Dark Matter Self-interactions and Small Scale Structure, Phys. Rept. 730, 1 (2018), arXiv:1705.02358 [hep-ph]

    2018 arXiv

  3. [11]

    Adhikari, A

    S. Adhikari, A. Banerjee, K. K. Boddy, F.-Y. Cyr- Racine, H. Desmond, C. Dvorkin, B. Jain, F. Kahlhoe- fer, M. Kaplinghat, A. Nierenberg, A. H. G. Peter, A. Robertson, J. Sakstein, and J. Zavala, Astrophysi- cal Tests of Dark Matter Self-Interactions, arXiv e-prints (2022), ar...

    2022

  4. [12]

    D. Yang, E. O. Nadler, H.-B. Yu, and Y.-M. Zhong, A parametric model for self-interacting dark matter halos, JCAP 02, 032, arXiv:2305.16176 [astro-ph.CO]

    arXiv

  5. [13]

    Hui, Wave Dark Matter, Ann

    L. Hui, Wave Dark Matter, Ann. Rev. Astron. Astrophys. 59, 247 (2021), arXiv:2101.11735 [astro-ph.CO]

    2021 arXiv

  6. [14]

    T. D. Yavetz, X. Li, and L. Hui, Construction of wave dark matter halos: Numerical algorithm and an- alytical constraints, Phys. Rev. D 105, 023512 (2022), arXiv:2109.06125 [astro-ph.CO]

    2022 arXiv

  7. [15]

    Kobyzev, S

    I. Kobyzev, S. J. Prince, and M. A. Brubaker, Normaliz- ing flows: An introduction and review of current meth- ods, IEEE transactions on pattern analysis and machine intelligence 43, 3964 (2020), arXiv:1908.09257 [stat.ML]

    2020 arXiv

  8. [16]

    G. M. Green and Y.-S. Ting, Deep Potential: Recover- ing the gravitational potential from a snapshot of phase space, in Machine Learning and the Physical Sciences, Workshop at the 34th Conference on Neural Information Processing Systems (NeurIPS) (2020) arXiv:2011.04673 [astro-ph.GA]

    2020 arXiv

  9. [17]

    S. H. Lim, K. Hayashi, S. Horigome, S. Matsumoto, and M. M. Nojiri, Generative adversarial network for identi- fying the dark matter distribution of a dwarf spheroidal galaxy, 4th Inter-experiment Machine Learning Work- shop (2020), contributed presentation

    2020

  10. [18]

    J. An, A. P. Naik, N. W. Evans, and C. Burrage, Charting galactic accelerations: when and how to ex- tract a unique potential from the distribution function, Monthly Notices of the Royal Astronomical Society 506, 5721 (2021), https://academic.oup.com/mnras/article- pdf/506/4/5...

    2021

  11. [19]

    A. P. Naik, J. An, C. Burrage, and N. W. Evans, Charting galactic accelerations - II. How to ’learn’ ac- celerations in the solar neighbourhood, Monthly Notices of the Royal Astronomical Society 511, 1609 (2022), arXiv:2112.07657 [astro-ph.GA]

    2022 arXiv

  12. [20]

    G. M. Green, Y.-S. Ting, and H. Kamdar, Deep Po- tential: Recovering the Gravitational Potential from a Snapshot of Phase Space, ApJ 942, 26 (2023), arXiv:2205.02244 [astro-ph.GA]

    2023 arXiv

  13. [21]

    M. R. Buckley, S. H. Lim, E. Putney, and D. Shih, Mea- suring Galactic dark matter through unsupervised ma- chine learning, Mon. Not. Roy. Astron. Soc. 521, 5100 (2023), arXiv:2205.01129 [astro-ph.GA]

    2023 arXiv

  14. [22]

    Brooks, h277-g14 simulation outputs (snapshots 24- 144 of 512), in The N-Body Shop (2020)

    A. Brooks, h277-g14 simulation outputs (snapshots 24- 144 of 512), in The N-Body Shop (2020)

    2020

  15. [23]

    Zolotov, A

    A. Zolotov, A. M. Brooks, B. Willman, F. Governato, A. Pontzen, C. Christensen, A. Dekel, T. Quinn, S. Shen, and J. Wadsley, Baryons Matter: Why Luminous Satel- lite Galaxies have Reduced Central Masses, ApJ 761, 71 (2012), arXiv:1207.0007 [astro-ph.CO]

    2012 arXiv

  16. [24]

    S. R. Loebman, ˇZ. Ivezi´ c, T. R. Quinn, F. Governato, A. M. Brooks, C. R. Christensen, and M. Juri´ c, Con- straints on the Shape of the Milky Way Dark Matter Halo from Jeans Equations Applied to Sloan Digital Sky Sur- vey Data, ApJ 758, L23 (2012), arXiv:1209.2708 [astro- ph.GA]

    2012 arXiv

  17. [25]

    S. H. Lim, E. Putney, M. R. Buckley, and D. Shih, Map- ping dark matter in the Milky Way using normalizing flows and Gaia DR3, JCAP 01, 021, arXiv:2305.13358 [astro-ph.GA]

  18. [26]

    Putney, D

    E. Putney, D. Shih, S. H. Lim, and M. R. Buckley, Map- ping Dark Matter Through the Dust of the Milky Way Part I: Dust Correction and Phase Space Density, (2024), arXiv:2412.14236 [astro-ph.GA]

    2024

  19. [27]

    Ullio and M

    P. Ullio and M. Valli, A critical reassessment of parti- cle dark matter limits from dwarf satellites, Journal of Cosmology and Astroparticle Physics 2016 (07), 025, arXiv:1603.07721 [astro-ph.GA]

    2016 arXiv

  20. [28]

    Hayashi, K

    K. Hayashi, K. Ichikawa, S. Matsumoto, M. Ibe, M. N. Ishigaki, and H. Sugai, Dark matter annihilation and decay from non-spherical dark halos in galactic dwarf satellites, Mon. Not. Roy. Astron. Soc. 461, 2914 (2016), arXiv:1603.08046 [astro-ph.GA]. 7

    2016 arXiv

  21. [29]

    Ichikawa, M

    K. Ichikawa, M. N. Ishigaki, S. Matsumoto, M. Ibe, H. Sugai, K. Hayashi, and S.-i. Horigome, Foreground effect on the J-factor estimation of classical dwarf spheroidal galaxies, Mon. Not. Roy. Astron. Soc. 468, 2884 (2017), arXiv:1608.01749 [astro-ph.GA]

    2017 arXiv

  22. [30]

    Ichikawa, S.-i

    K. Ichikawa, S.-i. Horigome, M. N. Ishigaki, S. Mat- sumoto, M. Ibe, H. Sugai, and K. Hayashi, Foreground effect on the J-factor estimation of ultrafaint dwarf spheroidal galaxies, Mon. Not. Roy. Astron. Soc. 479, 64 (2018), arXiv:1706.05481 [astro-ph.GA]

    2018 arXiv

  23. [31]

    Horigome, K

    S.-i. Horigome, K. Hayashi, M. Ibe, M. N. Ishi- gaki, S. Matsumoto, and H. Sugai, J-factor esti- mation of Draco, Sculptor and Ursa Minor dwarf spheroidal galaxies with the member/foreground mixture model, Mon. Not. Roy. Astron. Soc. 499, 3320 (2020), arXiv:2002.04866 [astro-ph.GA]

    2020 arXiv

  24. [32]

    F. I. Diakogiannis, G. F. Lewis, R. A. Ibata, M. Guglielmo, P. R. Kafle, M. I. Wilkinson, and C. Power, A novel JEAnS analysis of the Fornax dwarf using evolutionary algorithms: mass follows light with signs of an off-centre merger, MNRAS 470, 2034 (2017), arXiv:1705.05724 [as...

    2017 arXiv

  25. [33]

    F. I. Diakogiannis, G. F. Lewis, R. A. Ibata, M. Guglielmo, M. I. Wilkinson, and C. Power, Reliable mass calculation in spherical gravitating systems, MN- RAS 482, 3356 (2019), arXiv:1810.11375 [astro-ph.GA]

    2019 arXiv

  26. [34]

    J. I. Read and P. Steger, How to break the density- anisotropy degeneracy in spherical stellar systems, MN- RAS 471, 4541 (2017), arXiv:1701.04833 [astro-ph.GA]

    2017 arXiv

  27. [35]

    J. I. Read, M. G. Walker, and P. Steger, The case for a cold dark matter cusp in Draco, MNRAS 481, 860 (2018), arXiv:1805.06934 [astro-ph.GA]

    2018 arXiv

  28. [36]

    Genina, J

    A. Genina, J. I. Read, C. S. Frenk, S. Cole, A. Ben´ ıtez- Llambay, A. D. Ludlow, J. F. Navarro, K. A. Oman, and A. Robertson, To β or not to β: can higher or- der Jeans analysis break the mass-anisotropy degener- acy in simulated dwarfs?, MNRAS 498, 144 (2020), arXiv:1911.091...

    2020 arXiv

  29. [37]

    M. L. M. Collins, J. I. Read, R. A. Ibata, R. M. Rich, N. F. Martin, J. Pe˜ narrubia, S. C. Chapman, E. J. Tollerud, and D. R. Weisz, Andromeda XXI - a dwarf galaxy in a low-density dark matter halo, MNRAS 505, 5686 (2021), arXiv:2102.11890 [astro-ph.GA]

    2021 arXiv

  30. [38]

    R. T. Q. Chen, Y. Rubanova, J. Bettencourt, and D. K. Duvenaud, Neural ordinary differential equations, in Advances in Neural Information Processing Systems , Vol. 31, edited by S. Bengio, H. Wallach, H. Larochelle, K. Grauman, N. Cesa-Bianchi, and R. Garnett (Curran Associate...

    2018 arXiv

  31. [39]

    Grathwohl, R

    W. Grathwohl, R. T. Q. Chen, J. Bettencourt, and D. Duvenaud, FFJORD: Free-form continuous dynam- ics for scalable reversible generative models, in Inter- national Conference on Learning Representations (2019) arXiv:1810.01367 [cs.LG]

    2019 arXiv

  32. [40]

    K¨ ohler, L

    J. K¨ ohler, L. Klein, and F. No´ e, Equivariant flows: Ex- act likelihood generative learning for symmetric densities, in Proceedings of the 37th International Conference on Machine Learning, Proceedings of Machine Learning Re- search, Vol. 119, edited by H. D. III and A. Sin...

  33. [41]

    K¨ ohler, L

    J. K¨ ohler, L. Klein, and F. No´ e, Equivariant Flows: sam- pling configurations for multi-body systems with sym- metric energies, in Machine Learning and the Physi- cal Sciences, Workshop at the 33rd Conference on Neu- ral Information Processing Systems (NeurIPS) (2019) arXi...

    2019 arXiv

  34. [42]

    Kanwar, M

    G. Kanwar, M. S. Albergo, D. Boyda, K. Cranmer, D. C. Hackett, S. Racani` ere, D. J. Rezende, and P. E. Shanahan, Equivariant flow-based sampling for lattice gauge theory, Phys. Rev. Lett. 125, 121601 (2020), arXiv:2003.06413 [hep-lat]

    2020 arXiv

  35. [43]

    P. B. Eskridge, The Structure of the Sculptor Dwarf El- liptical Galaxy. I. The Radial Profile, AJ95, 1706 (1988)

    1988

  36. [44]

    P. B. Eskridge, The Structure of the Fornax Dwarf Ellip- tical Galaxy. I. The Radial Profile, AJ 96, 1352 (1988)

    1988

  37. [45]

    Irwin and D

    M. Irwin and D. Hatzidimitriou, Structural param- eters for the galactic dwarf spheroidals, Monthly Notices of the Royal Astronomical Society 277, 1354 (1995), https://academic.oup.com/mnras/article- pdf/277/4/1354/4300979/mnras277-1354.pdf

    1995

  38. [46]

    H. C. Plummer, On the problem of distribution in glob- ular star clusters, Monthly Notices of the Royal Astro- nomical Society 71, 460 (1911)

    1911

  39. [47]

    W. Jaffe, A simple model for the distribution of light in spherical galaxies, Monthly Notices of the Royal Astronomical Society 202, 995 (1983), https://academic.oup.com/mnras/article- pdf/202/4/995/18194452/mnras202-0995.pdf

    1983

  40. [48]

    Hernquist, An Analytical Model for Spherical Galaxies and Bulges, ApJ 356, 359 (1990)

    L. Hernquist, An Analytical Model for Spherical Galaxies and Bulges, ApJ 356, 359 (1990)

    1990

  41. [49]

    W. Dehnen, A family of potential–density pairs for spherical galaxies and bulges, Monthly No- tices of the Royal Astronomical Society 265, 250 (1993), https://academic.oup.com/mnras/article- pdf/265/1/250/2975628/mnras265-0250.pdf

    1993

  42. [50]

    J. L. S´ ersic, Influence of the atmospheric and instrumen- tal dispersion on the brightness distribution in a galaxy, Boletin de la Asociacion Argentina de Astronomia La Plata Argentina 6, 41 (1963)

    1963

  43. [51]

    de Vaucouleurs, Recherches sur les Nebuleuses Extra- galactiques, Annales d’Astrophysique 11, 247 (1948)

    G. de Vaucouleurs, Recherches sur les Nebuleuses Extra- galactiques, Annales d’Astrophysique 11, 247 (1948)

    1948

  44. [52]

    A. W. Graham, P. Erwin, I. Trujillo, and A. Asensio Ramos, A New Empirical Model for the Structural Anal- ysis of Early-Type Galaxies, and A Critical Review of the Nuker Model, AJ 125, 2951 (2003), arXiv:astro- ph/0306023 [astro-ph]

    2003

  45. [53]

    Goodfellow, J

    I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio, Generative adversarial nets, in Advances in Neural Infor- mation Processing Systems, Vol. 27, edited by Z. Ghahra- mani, M. Welling, C. Cortes, N. Lawrence, and K. Wein- b...

    2014 arXiv

  46. [54]

    Reed, Spherical & triaxial, in 3rd Gaia Challenge (2015)

    J. Reed, Spherical & triaxial, in 3rd Gaia Challenge (2015)

    2015

  47. [55]

    M. G. Walker and J. Pe˜ narrubia, A Method for Mea- suring (Slopes of) the Mass Profiles of Dwarf Spheroidal Galaxies, ApJ 742, 20 (2011), arXiv:1108.2404 [astro- ph.CO]

    2011 arXiv

  48. [56]

    J. Wolf, G. D. Martinez, J. S. Bullock, M. Kapling- hat, M. Geha, R. R. Munoz, J. D. Simon, and F. F. Avedo, Accurate masses for dispersion-supported galax- ies, Monthly Notices of the Royal Astronomical Society 406, 1220 (2010)

    2010

  49. [57]

    Nguyen, S

    T. Nguyen, S. Mishra-Sharma, R. Williams, and L. Necib, Uncovering dark matter density profiles in dwarf galaxies with graph neural networks, Phys. Rev. D 107, 043015 (2023), arXiv:2208.12825 [astro-ph.CO]. 8 TABLE I. Prior ranges and true values of the generalized Hernquist p...

    2023 arXiv

  50. [58]

    Nguyen, J

    T. Nguyen, J. Read, L. Necib, S. Mishra-Sharma, C.-A. Faucher-Gigu` ere, and A. Wetzel, Trial by FIRE: Probing the dark matter density profile of dwarf galaxies with GraphNPE, arXiv e-prints , arXiv:2503.03812 (2025), arXiv:2503.03812 [astro-ph.GA]

    2025 arXiv

  51. [59]

    Zhao, Analytical models for galactic nuclei, MNRAS 278, 488 (1996), arXiv:astro-ph/9509122 [astro-ph]

    H. Zhao, Analytical models for galactic nuclei, MNRAS 278, 488 (1996), arXiv:astro-ph/9509122 [astro-ph]

    1996 arXiv

  52. [60]

    Hendrycks and K

    D. Hendrycks and K. Gimpel, Gaussian error linear units (gelus), (2016), arXiv:1606.08415 [cs.LG]

    2016 arXiv

  53. [61]

    D. P. Kingma and J. Ba, Adam: A method for stochastic optimization, in the 3rd International Confer- ence for Learning Representations (ICLR 2015) (2014) arXiv:1412.6980 [cs.LG]. Appendix A: NonplumCoreIso Datasets Here we describe the simulation parameters of the NonplumCoreI...

    2014 arXiv

  54. [63]

    For other parameters, we use the default parameters in PyTorch

    with learning rate 0.001. For other parameters, we use the default parameters in PyTorch. The training pro- cess is repeated 20 times with different random seeds, and we report ensemble-averaged predictions to reduce ran- dom fluctuations. Stellar number density training-speci...

  55. [64]

    Initialize the cusp parameter c by randomly sam- pling the cusp scaling exponent− 3c c+1 from the uni- form distribution on (−1, 0]

  56. [65]

    The tran- sition scale rs in (8) is set to the 1D rule-of-thumb bandwidth for the standardized dataset, N− 1 5

    Fix the cusp parameter c and train only the net- work parameters; c will be updated later after fin- ishing the network parameter training. The tran- sition scale rs in (8) is set to the 1D rule-of-thumb bandwidth for the standardized dataset, N− 1 5

  57. [66]

    After training the network parameters, evaluate the cusp exponent d logn(r)/d logr at 300 random lo- cations sampled from a 3D Gaussian distribution centered at the galactic center, with standard de- viation N− 1 5 for each axis

  58. [67]

    Repeat steps 2-4

    Use the mean cusp exponent −¯γ = E [d logn(r)/d logr] as the next cusp scaling exponent, i.e., re-initialize c = ¯γ 3−¯γ for the next training iteration. Repeat steps 2-4. Each iteration with fixed c takes approximately 300-400 seconds on an NVIDIA L40S GPU with GPU utilizatio...

  59. [2020]

    5361–5370, arXiv:2006.02425 [stat.ML]

    pp. 5361–5370, arXiv:2006.02425 [stat.ML]

    2006 arXiv

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.