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  • Open Access

Q-balls, neural networks, and galaxy rotation curves

Alexandre M. Pombo1,*, Lorenzo Pizzuti2,†, and Alessandra di Giacomo2,‡

  • *Contact author: pombo@fzu.cz
  • †Contact author: lorenzo.pizzuti@unimib.it
  • ‡Contact author: a.digiacomo12@campus.unimib.it
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Phys. Rev. D 114, 083020 – Published 7 October, 2026

DOI: https://doi.org/10.1103/tz41-5qcp

Abstract

Can a dynamically robust (also known as stable) Q-ball reproduce the rotation curve of a disk galaxy? In an astrophysical environment, Q-balls are nontopological solitons that are transparent and only perceived by their gravitational effects. Traditionally, scalar Q-balls are modeled with a polynomial potential, but axionlike periodic potentials are also expected to support such solitonic configurations. In the presence of angular momentum, Q-balls acquire a toroidal structure with a central density void, qualitatively resembling the axially symmetric structure of disk galaxies. Motivated by this similarity, we explore whether rotating scalar Q-balls can reproduce the observed rotation curves of disk galaxies. In this work, we use a recently developed hybrid numerical framework that combines a high-accuracy pseudospectral method with a physics-informed neural network approach to construct both static and rotating Q-ball solutions. We assess the ability of this simplified Q-ball model to mimic the dark matter halos in galaxies by fitting the observed rotation curves of a sample of disk galaxies from the SPARC catalog. While Q-ball configurations are built considering a negligible self-gravity, we show their use as dark matter halos to be valid for the field parameters and halo masses relevant to disk galaxies. Our simplified model provides an overall good agreement with observational data when imposing physically motivated priors on the halo total mass, and a reasonable fit when compared to standard dark matter profiles such as the Navarro-Frenk-White; we have further found an average preferred range of the scalar field particle’s mass mϕ∼10−27  eV, in agreement with similar galactic-scale soliton solutions.

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References (55)

  1. V. C. Rubin, J. W. Kent Ford, and N. Thonnard, Extended rotation curves of high-luminosity spiral galaxies. IV.—Systematic dynamical properties, Sa through Sc, Astrophys. J. 225, L107 (1978).
  2. A. Bosma, 21-cm line studies of spiral galaxies. II. The distribution and kinematics of neutral hydrogen in spiral galaxies of various morphological types, Astron. J. 86, 1825 (1981).
  3. F. Lelli, S. S. McGaugh, and J. M. Schombert, Sparc: Mass models for 175 disk galaxies with spitzer photometry and accurate rotation curves, Astron. J. 152, 157 (2016).
  4. K. A. Oman, J. F. Navarro, A. Fattahi, C. S. Frenk, T. Sawala, S. D. M. White et al., The unexpected diversity of dwarf galaxy rotation curves, Mon. Not. R. Astron. Soc. 452, 3650 (2015).
  5. S.-H. Oh, D. A. Hunter, E. Brinks, B. G. Elmegreen, A. Schruba, F. Walter et al., High-resolution mass models of dwarf galaxies from LITTLE THINGS, Astrophys. J. 149, 180 (2015).
  6. I. S. Sands, P. F. Hopkins, X. Shen, M. Boylan-Kolchin, J. Bullock, C.-A. Fauch-er-Giguère et al., Confronting the diversity problem: The limits of galaxy rotation curves as a tool to understand dark matter profiles, Astrophys. J. 1000, 127 (2026).
  7. A. H. Nelson and P. R. Williams, Recent observations of the rotation of distant galaxies and the implication for dark matter, Astron. Astrophys. 687, A261 (2024).
  8. D. Jeong, H. S. Hwang, H. Chung, and Y. Yoon, Diverse rotation curves of galaxies in a simulated universe: The observed dependence on stellar mass and morphology reproduced, Astrophys. J. 982, 11 (2025).
  9. S. Tulin and H.-B. Yu, Dark matter self-interactions and small-scale structure, Phys. Rep. 730, 1 (2018).
  10. M. Milgrom, A modification of the newtonian dynamics as a possible alternative to the hidden mass hypothesis, Astrophys. J. 270, 365 (1983).
  11. T. Matsakos and A. Diaferio, Dynamics of galaxies and clusters in refracted gravity, arXiv:1603.04943.
  12. L. Pizzuti, F. Fantoccoli, V. Broccolato, A. Biviano, and A. Diaferio, Testing refracted gravity with the kinematics of galaxy clusters, Astron. Astrophys. 698, A83 (2025).
  13. B. Famaey, L. Pizzuti, and I. D. Saltas, Nature of the missing mass of galaxy clusters in MOND: The view from gravitational lensing, Phys. Rev. D 111, 123042 (2025).
  14. J. S. Bullock and M. Boylan-Kolchin, Small-scale challenges to the λcdm paradigm, Annu. Rev. Astron. Astrophys. 55, 343 (2017).
  15. D. J. E. Marsh and S. Hoof, Astrophysical searches and constraints, in The Search for Ultralight Bosonic Dark Matter (Springer International Publishing, New York, 2022), p. 73–122.
  16. R. Della Monica and I. de Martino, Bounding the mass of ultralight bosonic dark matter particles with the motion of the S2 star around SgrA*, Phys. Rev. D 108, L101303 (2023).
  17. L. Salasnich and A. Yakimenko, Collective excitations of self-gravitating ultralight dark matter cores, Phys. Dark Universe 49, 101973 (2025).
  18. P. Jetzer, Boson stars, Phys. Rep. 220, 163 (1992).
  19. B. Kleihaus, J. Kunz, and M. List, Rotating boson stars and Q-balls, Phys. Rev. D 72, 064002 (2005).
  20. C. A. Herdeiro, A. M. Pombo, and E. Radu, Asymptotically flat scalar, dirac and proca stars: Discrete vs continuous families of solutions, Phys. Lett. B 773, 654 (2017).
  21. A. M. Pombo, J. M. Oliveira, and N. M. Santos, Coupled scalar-proca soliton stars, Phys. Rev. D 108, 044044 (2023).
  22. R. Brito, V. Cardoso, C. A. Herdeiro, and E. Radu, Proca stars: Gravitating Bose–Einstein condensates of massive spin 1 particles, Phys. Lett. B 752, 291 (2016).
  23. Y. Shnir, Boson stars, in Modified and Quantum Gravity: From Theory to Experimental Searches on All Scales (Springer, New York, 2023), pp. 347–362.
  24. S. Coleman, Q-balls, Nucl. Phys. B262, 263 (1985).
  25. M. S. Volkov and E. Wöhnert, Spinning Q-balls, Phys. Rev. D 66, 085003 (2002).
  26. K. Lee, J. A. Stein-Schabes, R. Watkins, and L. M. Widrow, Gauged Q balls, Phys. Rev. D 39, 1665 (1989).
  27. V. Loiko, I. Perapechka, and Y. Shnir, Q-balls without a potential, Phys. Rev. D 98, 045018 (2018).
  28. M. Axenides, S. Komineas, L. Perivolaropoulos, and M. Floratos, Dynamics of nontopological solitons: Q balls, Phys. Rev. D 61, 085006 (2000).
  29. J. C. Mourelle and C. Adam, Galactic halos and rotating bosonic dark matter, Phys. Rev. D 110, 123001 (2024).
  30. J. C. Mourelle, N. Sanchis-Gual, and J. A. Font, Reproducing galactic rotation curves with a two-component bosonic dark matter model, Phys. Rev. D 112, 063054 (2025).
  31. D. Guerra, C. F. Macedo, and P. Pani, Axion boson stars, J. Cosmol. Astropart. Phys. 09 (2019) 061.
  32. J. F. Delgado, C. A. Herdeiro, and E. Radu, Rotating axion boson stars, J. Cosmol. Astropart. Phys. 06 (2020) 037.
  33. A. M. Pombo and I. Saltas, A sun-like star orbiting a boson star, Mon. Not. R. Astron. Soc. 524, 4083 (2023).
  34. C. A. Herdeiro, A. M. Pombo, E. Radu, P. V. Cunha, and N. Sanchis-Gual, The imitation game: Proca stars that can mimic the Schwarzschild shadow, J. Cosmol. Astropart. Phys. 04 (2021) 051.
  35. G. Derrick, Comments on nonlinear wave equations as models for elementary particles, J. Math. Phys. (N.Y.) 5, 1252 (1964).
  36. A. M. Pombo and L. Pizzuti, Virial identities across the spacetime, Phys. Rev. D 111, 084008 (2025).
  37. J. M. Oliveira and A. M. Pombo, A convenient gauge for virial identities in axial symmetry, Phys. Lett. B 837, 137646 (2023).
  38. C. A. Herdeiro, J. M. Oliveira, A. M. Pombo, and E. Radu, Deconstructing scaling virial identities in general relativity: Spherical symmetry and beyond, Phys. Rev. D 106, 024054 (2022).
  39. C. A. Herdeiro, J. M. Oliveira, A. M. Pombo, and E. Radu, Virial identities in relativistic gravity: 1D effective actions and the role of boundary terms, Phys. Rev. D 104, 104051 (2021).
  40. J. L. Blazquez-Salcedo, F. S. Khoo, J. Kunz, and L. M. Gonzalez-Romero, Quasinormal modes of Kerr black holes using a spectral decomposition of the metric perturbations, Phys. Rev. D 109, 064028 (2024).
  41. R. Luna, J. Calderón Bustillo, J. J. Seoane Martínez, A. Torres-Forne, and J. A. Font, Solving the Teukolsky equation with physics-informed neural networks, Phys. Rev. D 107, 064025 (2023).
  42. A. M. Pombo, Black holes and solitonic objects with bosonic fields, arXiv:2503.20444.
  43. J. Heeck, A. Rajaraman, R. Riley, and C. B. Verhaaren, Understanding Q-balls beyond the thin-wall limit, Phys. Rev. D 103, 045008 (2021).
  44. M. I. Tsumagari, E. J. Copeland, and P. M. Saffin, Some stationary properties of a Q-ball in arbitrary space dimensions, Phys. Rev. D 78, 065021 (2008).
  45. M. I. Tsumagari, The physics of Q-balls, arXiv:0910.3845.
  46. J. Binney and S. Tremaine, Galactic Dynamics, 2nd ed. (Princeton University Press, Princeton, NJ, 2008).
  47. K. Sheth, M. W. Regan, J. L. Hinz, A. Gil de Paz et al., The spitzer survey of stellar structure in galaxies (sg4), Publ. Astron. Soc. Pac. 122, 1397 (2010).
  48. J. M. Schombert, Archangel: A multi-purpose photometry package, Astrophysics Source Code Library (2011) ascl:1107.011.
  49. S. Casertano, Rotation curve of the edge-on spiral galaxy NGC 5907: Mass distribution and the rotation of the outer disk, Mon. Not. R. Astron. Soc. 203, 735 (1983).
  50. J. M. Schombert, S. S. McGaugh, and F. Lelli, The mass-to-light ratio of spiral galaxies, Astron. J. 147, 34 (2014).
  51. S. S. McGaugh, J. M. Schombert, and F. Lelli, Stellar population synthesis models and the mass-to-light ratio of spiral galaxies, Astrophys. J. 788, 32 (2014).
  52. J. F. Navarro, C. S. Frenk, and S. D. M. White, The structure of cold dark matter Halos, Astrophys. J. 462, 563 (1996).
  53. J. N. López-Sánchez, E. Munive-Villa, C. Skordis, and F. R. Urban, Scaling relations and tidal disruption in spin s ultralight dark matter models, Mon. Not. R. Astron. Soc. 543, 4092 (2025).
  54. S. L. Zoutendijk, J. Brinchmann, N. F. Bouché, M. den Brok, D. Krajnović, K. Kuijken, M. V. Maseda, and J. Schaye, The MUSE-Faint survey. II. The dark-matter density profile of the ultrafaint dwarf galaxy Eridanus 2, Astron. Astrophys. 651, A80 (2021).
  55. A. Bañares-Hernández, A. Castillo, J. Martin Camalich, and G. Iorio, Confronting fuzzy dark matter with the rotation curves of nearby dwarf irregular galaxies, Astron. Astrophys. 676, A63 (2023).

Outline

Information

Phys. Rev. D 114, 083020– Published 7 October, 2026
  • Received 9 February 2026
  • Accepted 24 September 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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