The governing equations of fuzzy dark matter are deceptively compact. The model is described by the Schrödinger–Poisson system, in which a complex wave function evolves under the influence of its own self-gravity. Despite their quantum pedigree, these equations admit a fluid interpretation through the Madelung transformation, which recasts the wave function in terms of a density and a velocity field. In this fluid picture, the familiar equations of hydrodynamics acquire an additional term, the quantum pressure, which counteracts gravity on small scales and introduces a characteristic Jeans scale. Structures larger than this scale grow much as they would in the standard cold dark matter scenario, while smaller structures are suppressed, because the uncertainty principle prevents the bosons from being localized into arbitrarily tight clumps. This suppression of small-scale structure is one of the model’s most distinctive and testable predictions, and it manifests as a sharp cutoff in the linear matter power spectrum at a wavenumber that scales with the square root of the boson mass.

Simulating this quantum fluid is dramatically harder than simulating cold dark matter. Wave-based schemes, which evolve the wave function directly, must resolve not only the de Broglie wavelength but also its rapid oscillations in time. The time-step constraints scale with the square of the spatial resolution, a diffusion-like requirement that makes high-resolution simulations extraordinarily expensive. The review details how the computational cost of resolving density granulation within a halo scales as the fourth power of the boson mass parameter and as the 8/3 power of halo mass, meaning that simulations of massive halos with heavier bosons quickly become infeasible. This is the central reason why most genuine wave-based cosmological simulations to date have been restricted to small boxes of a few megaparsecs, or to relatively light boson masses, while heavier candidates and larger volumes remain out of reach.

Among wave-based methods, the pseudo-spectral approach reigns supreme in accuracy. The global Fourier method applies a fast Fourier transform to the wave function, multiplies by the kinetic operator in momentum space, and transforms back, achieving spectral convergence with errors that decay faster than any algebraic rate as resolution increases. The method is unconditionally stable and conserves mass to machine precision, making it the algorithm of choice for codes such as GAMER, AREPO, PyUltraLight and SPoS. Its Achilles heel is that the discrete Fourier transform assumes a uniform grid with periodic boundaries, which is incompatible with adaptive mesh refinement, the technique that makes cosmological simulations affordable by concentrating resolution only where it is needed. This limitation has spurred the development of alternatives, including finite-difference schemes that work on refined grids but converge far more slowly, and a newer local pseudo-spectral technique based on Fourier continuations with Gram polynomials, which achieves eleventh-order accuracy on non-periodic grids and has been implemented in the GAMER code.

Fluid-based methods offer a compelling escape from the resolution bottleneck. Because the density, velocity and phase fields are smooth in the vast low-density regions outside halos, fluid schemes do not need to resolve the de Broglie wavelength there, allowing much larger simulation volumes. Smoothed particle hydrodynamics, implemented in codes such as AX-GADGET, and meshless finite-volume methods in the GIZMO code both exploit this advantage, and the finite-volume formulation conserves mass and momentum to machine precision. But fluid approaches have a fundamental weakness: at density nodes created by destructive wave interference, the velocity and quantum potential diverge, and the phase field becomes discontinuous. These singular points, which correspond to quantized vortices threading the halos, are notoriously difficult for any fluid scheme to capture, and the review documents persistent discrepancies between SPH simulations and wave-based calculations, including disagreements over whether soliton cores and the soliton–halo relation are recovered correctly.

The most promising path forward may be hybrid schemes that combine the strengths of both formulations. In the approach developed by Schwabe and Niemeyer and implemented in the AxioNyx code, collisionless N-body particles carry wave packets on coarse levels, and the full wave function is reconstructed on fine levels using a Gaussian beam method that captures statistically correct interference patterns. A newer fully grid-based hybrid scheme in GAMER, described by Kunkel and collaborators, solves the Hamilton–Jacobi–Madelung fluid equations on coarse levels and switches to the wave formulation on refined levels, with careful phase unwrapping at the fluid–wave interfaces. These hybrid methods, coupled with adaptive mesh refinement, have enabled cosmological zoom-in simulations of individual halos down to the present day, something that pure wave-based schemes could not previously achieve, and they open the door to simulating more massive halos and heavier boson masses than ever before.

The distinctive phenomenology of fuzzy dark matter gives simulations their scientific payoff. Each virialized halo hosts a dense, stable soliton core, the ground-state solution of the halo potential, surrounded by an outer profile that closely resembles the standard Navarro–Frenk–White form. The soliton density profile is redshift-independent and obeys an elegant scaling symmetry, and its properties are linked to the host halo through a debated soliton–halo relation that recent work interprets as thermal equilibrium between the core and the surrounding granules. Beyond the soliton, the halo is permeated by stochastically fluctuating density granules, roughly a quarter of the local de Broglie wavelength in size, generated by constructive and destructive interference. These granules scatter stars and gas, dynamically heating dwarf galaxies, stellar streams and galactic disks, and their effects are now being used to place some of the tightest constraints on the boson mass, with recent analyses of ultrafaint dwarfs suggesting the mass must exceed roughly 2.2 x 10^-21 electronvolts.

Simulations also reveal subtler pitfalls that can silently corrupt results. Spurious halos, seeded by numerical noise along filaments, can form below the half-mode mass in both N-body and genuine fuzzy dark matter simulations, mimicking real objects and potentially biasing constraints derived from the halo mass function. The review shows that the global Fourier method appears largely immune to this artificial fragmentation, while hybrid and N-body schemes produce suspiciously regular clumps along filaments that lack counterparts across different numerical setups. Equally treacherous is insufficient resolution of the de Broglie wavelength in low-density regions, which underestimates flow velocities and delays structure formation, and inadequate resolution within halos, which distorts the quantum pressure and can drive unphysical halo contraction. The review demonstrates that even when a simulated halo’s mass and central soliton profile look correct, the outer profile can still be wrong, a sobering warning that agreement with theory is necessary but not sufficient evidence of numerical accuracy.

To help the community navigate these hazards, the review provides a test bench of standard problems, from Gaussian wave packets and vortex pairs to the Jeans instability, isolated solitons, isolated halos and full cosmological boxes, and it makes publicly available initial condition files for isolated-halo and cosmological simulations so that different codes can be compared on equal footing. Eigenmode methods, which construct halos as superpositions of bound states of a static potential, offer yet another efficient tool for studying interference phenomena without solving the full nonlinear system, while collisionless N-body simulations with fuzzy initial conditions remain valuable for probing large-scale structure and checking convergence. Looking ahead, the review notes that machine learning approaches, such as physics-informed neural networks, may eventually accelerate these calculations, and that the algorithms can be extended to self-interacting, vector and multi-component variants of the model. What remains clear is that simulating fuzzy dark matter has matured into a rich computational discipline in its own right, one whose progress will determine whether this quantum alternative to cold dark matter survives the confrontation with the universe it seeks to explain.

Subject of Research: Numerical simulation methods for fuzzy dark matter, an ultralight bosonic dark matter model exhibiting wave phenomena on galactic scales

Article Title: Fuzzy dark matter simulations

Article References: Schive, H.-Y. (2026). Fuzzy dark matter simulations. Living Reviews in Computational Astrophysics, 12(1), Article 1. https://doi.org/10.1007/s41115-026-00027-5

Image Credits: AI Generated

DOI: 10.1007/s41115-026-00027-5

Keywords: fuzzy dark matter, ultralight bosons, Schrödinger–Poisson equations, soliton cores, density granulation, adaptive mesh refinement, pseudo-spectral methods, quantum pressure, cosmological simulations, halo mass function, vortices, computational astrophysics