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References

This page collects the scientific literature and software packages cited in the WaveDM.jl paper and documentation. It is meant as a quick reference for users who want to follow up on the theoretical background or compare with alternative implementations.

1. Wave Dark Matter Theory

  • Hu, W., Barkana, R. & Gruzberg, A. (2000). Fuzzy cold dark matter: the wave properties of ultralight particles. Phys. Rev. Lett. 85, 1158. — hu2000fuzzy. The original proposal of fuzzy / wave CDM as a way to address small-scale CDM challenges.
  • Hui, L. (2021). Wave dark matter. Annu. Rev. Astron. Astrophys. 59, 247. — hui2021wavedark. A modern review of wave CDM: de Broglie scale, solitons, granules, observations.
  • Hui, L., Ostriker, J. P., Tremaine, S. & Witten, E. (2017). Ultralight scalars as cosmological dark matter. Phys. Rev. D 95, 043541. — hui2017ultralight. Defines the quantum pressure and "quantum energy" used in virial diagnostics.
  • Chavanis, P.-H. (2019). Predictive model of BEC dark matter halos with a solitonic core and a quantum pressure bulge. Phys. Rev. D 100, 083022. — chavanis2019predictive. Critical-temperature estimate for ultralight bosons.
  • Kagan, Yu., Muryshev, A. E. & Shlyapnikov, G. V. (1997). Evolution of correlation properties and appearance of interference in the course of Bose-Einstein condensation. Phys. Rev. Lett. 81, 933. — kagan1997evolutioncorrelation. Justifies the classical-field GPE in the highly-degenerate regime.
  • Dong, X.-B. et al. (2025). The promise of wave dark matter. (in prep.). — dong2025promise. Discussion of the excited-condensate component and its observational consequences (JWST).
  • Liu, W.-J. et al. (2025). Warm fuzzy dark matter. (in prep.). — liu2025warmfuzzy. Wave warm dark matter, an alternative to the cold case.
  • Chiang, B. T., Ostriker, J. P. & Schive, H.-Y. (2021). Soliton oscillations and revised constraints from the soliton–halo relation and from pulsar timing. Mon. Not. R. Astron. Soc. 508, 3945. — chiang2021soliton. Ground-state oscillation period $\tau_{00}$.

2. Schrödinger–Poisson Equation

  • Pitaevskii, L. P. & Stringari, S. (2016). Bose–Einstein Condensation and Superfluidity. Oxford University Press. — pitaevskii2016boseeinstein. Background on the GPE.
  • Liao, S. et al. (2025). Deciphering the soliton–halo relation and the quantum pressure in fuzzy dark matter halos. (in prep.). — liao2025decipheringsolitonhalo. Discusses the velocity-dispersion QSS diagnostic and the "effective thermal energy" terminology.
  • Mocz, P. et al. (2017). Galaxy formation with BECDM — I. Turbulence and relaxation of idealized halos. Mon. Not. R. Astron. Soc. 471, 4559. — mocz2017galaxy. Defines the "gradient energy" $K_\rho$ notation used in the virial-diagnostic literature.

3. Numerical Methods

  • Edwards, F., Kendall, E., Hotchkiss, S. & Sherwin, R. (2018). PyUltraLight: a pseudo-spectral solver for ultralight dark matter dynamics. (ascl:1808.008). — edwards2018pyultralight. The original pseudo-spectral SPE solver in Python; WaveDM.jl follows the same dimensionless-unit convention.
  • Glennon, N., Musoke, N., Chatterjee, A. & Nikoleyko, D. (2021). Modifying SPH for simulations of ultralight bosonic dark matter. Phys. Rev. D 105, 123540. — glennon2021modifying. Mocks the same dimensionless units.
  • Schwabe, B., Gosenca, M., Behrendt, C., Easther, R. & McDonald, J. (2016). Simulations of ultralight axion-like dark matter. Phys. Rev. D 95, 083513. — schwabe2016simulations. CFL timestep criterion.
  • May, S. & Springel, V. (2021). Structure formation in large-volume cosmological simulations of fuzzy dark matter. Mon. Not. R. Astron. Soc. 506, 2603. — may2021structure. Moving-mesh SPE.
  • May, S., Springel, V. & White, S. D. M. (2023). The halo structure of fuzzy dark matter. (in prep.). — may2023halo. Velocity resolution criterion.
  • Li, X., Hui, L. & Yavetz, T. (2019). Numerical challenge of the fuzzy dark matter model. Phys. Rev. D 103, 023508. — li2019numerical. Resolution criteria; de Broglie-wavelength requirement.
  • Schive, H.-Y., Chiueh, T. & Broadhurst, T. (2014, 2026 update). Cosmic structure as the quantum interference of a coherent dark wave. Nature Physics 10, 496; updated review preprint 2026. — schive2014cosmicstructure, schive2026fuzzydark. The reference paper for the SPE + AMR approach and a comprehensive review of methodologies.
  • James, R. A. (1977). The solution of Poisson's equation for isolated source distributions. J. Comput. Phys. 25, 71. — james1977solutionpoissons. The four-step algorithm implemented in PhysicalFFT.jl for isolated potentials; the current WaveDM.jl driver uses an isolated tree solve for boundary = Vacuum().
  • Dutta Chowdhury, D., van den Bosch, F. C., Robles, V. H., van Dokkum, P., Schive, H.-Y., Chiueh, T. & Broadhurst, T. (2021). On the random motion of the soliton in fuzzy dark matter halos. Astrophys. J. 916, 27. — duttachowdhury2021random. Box-size and resolution discussion for isolated halos.

4. Self-Interaction & Nonlinear Schrödinger

  • Levkov, D., Panin, A. & Tkachev, I. (2018). Gravitational Bose-Einstein condensation in the kinetic regime. Phys. Rev. Lett. 121, 151301. — levkov2018gravitational.
  • Nori, M. & Baldi, M. (2018). AX-GADGET: a new code for cosmological simulations of fuzzy dark matter and axion models. Mon. Not. R. Astron. Soc. 478, 3935. — nori2018axgadget.

5. Hydrodynamic / Madelung-based Methods

  • Mocz, P. & Succi, S. (2015). Numerical solution of the nonlinear Schrödinger equation using smoothed-particle hydrodynamics. Phys. Rev. E 91, 053304. — mocz2015numerical.
  • Veltmaat, J., Niemeyer, J. C. & Schwabe, B. (2016). Formation and structure of ultralight bosonic dark matter halos. Phys. Rev. D 94, 123523. — veltmaat2016cosmological.
  • Veltmaat, J., Schwabe, B. & Niemeyer, J. C. (2018). Formation of ultralight bosonic dark matter halos through a real scalar field. Phys. Rev. D 98, 043509. — veltmaat2018formation.
  • Zhang, J., Liu, H. & Chu, M.-C. (2018). Axion-Gadget: a smoothed-particle-hydrodynamics code for ultralight dark matter. (preprint). — zhang2018ultralight.
  • Mocz, P. et al. (2017). Galaxy formation with BECDM — I. Turbulence and relaxation of idealized halos. Mon. Not. R. Astron. Soc. 471, 4559. — mocz2017galaxy.
  • Mocz, P. et al. (2020). Galaxy formation with BECDM — II. Cosmic filaments and halos. Mon. Not. R. Astron. Soc. 494, 2027. — mocz2020galaxy.
  • Mocz, P. et al. (2025). jaxion: a high-performance code for ultralight dark matter. (preprint). — mocz2025jaxion.
  • Schwabe, B. & Niemeyer, J. C. (2020). Simulations of solitonic core plus fuzzy dark halo formation. Phys. Rev. Lett. 124, 201301. — schwabe2020simulating.
  • Schwabe, B., Gosenca, M., Behrendt, C., Easther, R. & McDonald, J. (2022). Deep learning for freckles in BEC dark matter. Phys. Rev. D 105, 123502. — schwabe2022deep.
  • Lague, A., Schwabe, B., Clough, K. & Niemeyer, J. C. (2021). Evolving fuzzy dark matter perturbations. Phys. Rev. D 104, 083536. — lague2021evolving.
  • Hopkins, P. F. et al. (2019). Stable hierarchical solar system and ultra-light dark matter. Mon. Not. R. Astron. Soc. 488, 4168. — hopkins2019stable.
  • Mina, M., Mota, D. F. & Winther, H. A. (2020). SCALAR: an AMR code for fuzzy dark matter. (preprint). — mina2020scalar.
  • Nori, M., Mocz, P., Davé, R. & Baldi, M. (2024). Fuzzy-Gasoline: a multi-fluid SPH code for ultralight bosonic dark matter. (preprint). — nori2024fuzzy.
  • Musoke, N., Hotchkiss, S. & Easther, R. (2024). UltraDark.jl: a Julia package for ultralight dark matter dynamics. J. Open Source Softw. — musoke2024ultradarkjl. Closest Julia-language SPE cousin.
  • Kunkel, S. et al. (2025). Hybrid CPU/GPU and MPI parallelization of the GAMER code. Astrophys. J. Suppl. Ser. — kunkel2025hybrid. The MPI/AMR SPE implementation in GAMER.
  • Hlozek, R. et al. (2015). A search for ultralight axions using the CMB. Phys. Rev. D 91, 103512. — hlozek2015search. AxionCAMB.

The full list with method, boundary, capabilities is reproduced in Table 1 of the accompanying paper (Meng & Dong 2026).

7. Observational Data & Galaxy Models

  • Battaglia, G., Taibi, S., Thomas, G. F. & Fritz, T. K. (2022). Gaia early DR3 systemic motions of Local Group dwarf galaxies. Astron. Astrophys. 657, A89. — battaglia2022gaia. Source catalogue for AstroIC.load_data_MW_satellites().
  • Hayashi, K. et al. (2023). Dark matter halos of dwarf galaxies from cosmological simulations. (preprint). — hayashi2023dark. Crater II gNFW fit used in the case study.
  • Zhu, Y. et al. (2023). How baryonic processes affect the Milky Way mass model. Mon. Not. R. Astron. Soc. — zhu2023how. Multi-component MW mass model (bulge, disk, gas, halo).

8. QSS / Quasi-Stationary State Theory

  • Lynden-Bell, D. (1967). Statistical mechanics of violent relaxation in stellar systems. Mon. Not. R. Astron. Soc. 136, 101. — lynden-bell1967statisticalmechanics.
  • Benetti, F. P. C. & Marcos, B. (2014). Non-equilibrium phase transitions in a 3D self-gravitating system. (preprint). — benetti2014nonequilibrium.
  • Klessen, R. S. (2000). Gravitational turbulence. Habilitation thesis. — klessen2000gravitational.
  • Berman, R. H. & Tajima, T. (2012). Turbulence in self-gravitating systems. (preprint). — berman2012turbulence.
  • Moon, S. et al. (2024). Theory of soliton–halo co-evolution in fuzzy dark matter. (preprint). — moon2024theory.
  • Chandrasekhar, S. (1953). Problems of stability in gravitational systems. Rev. Mod. Phys. — chandrasekhar1953problems.
  • Woo, T.-P. & Chiueh, T. (2009). High-resolution simulation on structure formation with extremely light bosonic dark matter. Astrophys. J. 697, 850. — woo2009highresolution.

9. Software References

9.1 Key dependencies

  • Bezanson, J., Edelman, A., Karpinski, S. & Shah, V. B. (2017). Julia: a fresh approach to numerical computing. SIAM Review 59, 65. — The Julia language.
  • Frigo, M. & Johnson, S. G. (2005). The design and implementation of FFTW3. Proc. IEEE 93, 216. — FFTW.jl.
  • Besard, T., Foket, C. & De Sutter, B. (2019). Effective extensible programming: unleashing Julia on GPUs. IEEE Trans. Parallel Distrib. Syst. — CUDA.jl.
  • Byrne, S., Wilcox, L. C. & Hamrud, M. (2021). MPI.jl: Julia bindings for the Message Passing Interface. (preprint). — MPI.jl. (Not currently used; planned for a future release.)
  • Danisch, S. & Krumbiegel, J. (2021). Makie.jl: flexible high-performance data visualization for Julia. J. Open Source Softw. — danisch2021makiejlflexible. GLMakie.jl, CairoMakie.jl.
  • Darema, F. et al. (1988). A single-program-multiple-data computational model for EPEX/FORTRAN. Parallel Comput. — darema1988singleprogrammultipledatacomputational. The SPMD paradigm underlying WaveDM.jl's Distributed.jl integration.
  • Springel, V. (2005). The cosmological simulation code GADGET-2. Mon. Not. R. Astron. Soc. 364, 1105. — springel2005cosmological. Tree / SPH reference.
  • Springel, V., Yoshida, N. & White, S. D. M. (2001). GADGET: a simulation environment for cosmological and galactic dynamics. New Astron. 6, 79. — springel2001gadget.
  • Gropp, W., Lusk, E. & Skjellum, A. (1999). Using MPI: portable parallel programming with the message-passing interface. MIT Press. — gropp1999usingmpi.

9.2 JuliaAstroSim Ecosystem

The full ecosystem is at https://github.com/JuliaAstroSim.

10. WaveDM.jl Citation

If you use WaveDM.jl in your research, please cite it as:

@article{meng2026wavedmjladaptablea,
  title = {{{WaveDM}}.Jl: {{An Adaptable Simulation Framework}} for {{Dynamics}} of {{Baryonic}} and {{Wave Dark Matter}} on {{Galaxy Scales}}},
  shorttitle = {{{WaveDM}}.Jl},
  author = {Meng, Runyu and Dong, Xiaobo},
  year = 2026,
  journal = {Research in Astronomy and Astrophysics},
  issn = {1674-4527},
  doi = {10.1088/1674-4527/ae9042},
  url = {http://iopscience.iop.org/article/10.1088/1674-4527/ae9042},
  eprint = {2606.25026},
  archiveprefix = {arXiv},
  primaryclass = {astro-ph.GA},
  urldate = {2026-07-26}
}

Publications built on this code base:

@article{meng2026wavecold,
  title = {Wave {{Cold Dark Matter Can Reproduce}} the {{Radial Acceleration Relation}} in {{X}}-Ray {{Elliptical Galaxies}}},
  author = {Meng, Runyu and Dong, Xiaobo},
  year = 2026,
  journal = {Research in Astronomy and Astrophysics},
  issn = {1674-4527},
  doi = {10.1088/1674-4527/aeac08},
  url = {http://iopscience.iop.org/article/10.1088/1674-4527/aeac08},
  eprint = {2610.18969},
  archiveprefix = {arXiv},
  primaryclass = {astro-ph.GA},
  urldate = {2026-09-24}
}

11. Online Resources