Standard, thermally generated dark matter remains firmly undetected in laboratory searches for weakly interacting massive particles (WIMPs; ref. 13). Non-thermal bosonic fields, particularly scalar fields, provide another well-motivated class of dark matter, formed in a non-relativistic, low-momentum state as a cold Bose–Einstein condensate (BEC), and increasingly motivated by extensions of the Standard Model of particle physics and to the mechanism driving the universal expansion14. The field in this context can be described by a coherent wave function ψ with an interference pattern determining the distribution of dark matter, which we term ψDM. Axions are long-standing CDM candidates of this form, and higher-dimensional theories motivate an ‘axiverse’, where a discrete mass spectrum of axion-like particles spans many decades, possibly affecting cosmic structure15.

The distribution of ψDM mimics particle CDM on large scales16,17, and hence distinguishing between CDM and cold, wavelike ψDM is best made on small scales owing to the additional quantum stress10,11,12,17. Dwarf spheroidal (dSph) galaxies are the smallest and most common class of galaxy with internal motions dominated by dark matter. Their basic properties are very hard to explain with standard CDM, including the surprising uniformity of their central masses, M(<300 pc) 107 M, where M is the solar mass, and shallow density profiles1,2,3,4,5. In contrast, galaxies predicted by CDM extend to much lower masses, well below the observed dwarf galaxies, with steeper, singular mass profiles6,7,8,9. Adjustments to standard CDM addressing these difficulties consider particle collisions18, or warm dark matter (WDM; ref. 19). WDM can be tuned to suppress small-scale structures, but does not provide large enough flat cores20. Collisional CDM can be adjusted to generate flat cores, but cannot suppress low-mass galaxies without resorting to other baryonic physics21. Better agreement is expected for ψDM because the uncertainty principle counters gravity below a Jeans scale, simultaneously suppressing small-scale structures and limiting the central density of collapsed haloes10,11,12.

Detailed examination of structure formation with ψDM is therefore highly desirable, but, unlike the extensive N-body investigation of standard CDM, no sufficiently high resolution simulations of ψDM have been attempted. The wave mechanics of ψDM can be described by Schrödinger’s equation, coupled to gravity by means of Poisson’s equation16 with negligible microscopic self-interaction. The dynamics here differs from collisionless particle CDM by a new form of stress tensor from quantum uncertainty, giving rise to a co-moving Jeans length, during the matter-dominated epoch17. The insensitivity of λJ to redshift, z, generates a sharp cutoff mass below which structures are suppressed. Cosmological simulations in this context turn out to be much more challenging than standard N-body simulations, as the highest frequency oscillations, ω, given approximately by the matter wave dispersion relation, where λ is the wavelength, occur on the smallest scales, requiring very fine temporal resolution even for moderate spatial resolution (Supplementary Fig. 1). In this work, we optimize an adaptive-mesh-refinement (AMR) scheme, with graphic processing unit acceleration, improving performance by almost two orders of magnitude22 (see Supplementary Section 1 for details).

Figure 1 demonstrates that despite the completely different calculations employed, the pattern of filaments and voids generated by a conventional N-body particle ΛCDM simulation is remarkably indistinguishable from the wavelike ΛψDM for the same linear power spectrum (Supplementary Fig. 3). Here Λ represents the cosmological constant. This agreement is desirable given the success of standard ΛCDM in describing the statistics of large-scale structure. To examine the wave nature that distinguishes ψDM from CDM on small scales, we re-simulate with a very high maximum resolution of 60 pc for a 2 Mpc co-moving box, so that the densest objects formed of 300 pc size are well resolved with 103 grids. A slice through this box is shown in Fig. 2, revealing fine interference fringes defining long filaments, with tangential fringes near the boundaries of virialized objects, where the de Broglie wavelengths depend on the local velocity of matter. An unexpected feature of our ψDM simulations is the generation of prominent dense coherent standing waves of dark matter in the centre of every gravitational bound object, forming a flat core with a sharp boundary (Figs 2 and 3). These dark matter cores grow as material is accreted and are surrounded by virialized haloes of material with fine-scale, large-amplitude cellular interference, which continuously fluctuate in density and velocity, generating quantum and turbulent pressure support against gravity.

Figure 1: Comparison of cosmological large-scale structures formed by standard CDM and by wavelike dark matter, ψDM.
figure 1

a, Structure created by evolving a single coherent wave function for ΛψDM calculated on adaptive-mesh-refinement grids. b, Structure simulated with a standard ΛCDM N-body code GADGET-2 (ref. 34) for the same cosmological parameters, with the high-k modes of the linear power spectrum intentionally suppressed in a way similar to the ψDM model to highlight the comparison of large-scale features. This comparison clearly demonstrates that the large-scale distribution of filaments and voids is indistinguishable between our model and ΛCDM (which has been successful in describing the observed large-scale structure). ψDM arises from the low-momentum state of the condensate so that it is equivalent to collisionless CDM well above the Jeans scale.

Figure 2: A slice of the density field of the ψDM simulation on various scales at z = 0.1.
figure 2

This scaled sequence (each of thickness 60 pc) shows how quantum interference patterns can be clearly seen everywhere from the large-scale filaments, tangential fringes near the virial boundaries, to the granular structure inside the haloes. Distinct solitonic cores with radii 0.3–1.6 kpc are found within collapsed haloes (which have virial masses Mvir 109–1011 M). The density shown here spans over nine orders of magnitude, from 10−1 to 108 (normalized to the cosmic mean density). The colour map scales logarithmically, with cyan corresponding to density 10.

Figure 3: Radial density profiles of haloes formed in the ψDM model.
figure 3

Dashed lines with various symbols show six examples of the halo profiles normalized to the cosmic mean density. All haloes are found to possess a distinct inner core fitted extremely well by the soliton solution (solid lines). A detailed soliton fit for the largest halo is inset, where the error is the root-mean-square scatter of density in each radial bin. A Navarro–Frenk–White (NFW) profile representing standard CDM is also shown for comparison (black dot-dashed line, with a very large scale radius of 10 kpc), which fits well the profiles outside the cores. The yellow hatched area indicates the ρ300 of the dSph satellites around the Milky Way3,24, which is consistent with the majority of galaxy haloes formed in the ψDM simulations.

The central density profiles of all our collapsed cores fit well the stable soliton solution of the Schrödinger–Poisson equation, as shown in Fig. 3 (see also Supplementary Section 2 and Figs 2 and 4). On the other hand, except for the lightest halo, which has just formed and is not yet virialized, the outer profiles of other haloes possess a steepening logarithmic slope, similar to the Navarro–Frenk–White (NFW) profile23 of standard CDM. These solitonic cores, which are gravitationally self-bound and appear as additional mass clumps superposed on the NFW profile, are clearly distinct from the cores formed by WDM and collisional CDM, which truncate the NFW cuspy inner profile at lower values and require an external halo for confinement. The radius of the soliton scales inversely with mass, such that the widest cores are the least massive and are hosted by the least massive galaxies. Eighty percent of the haloes in the simulation have an average density within 300 pc (defined as ρ300) in the range 5.3 × 10−3–6.1 × 10−1 M/pc3, consistent with the dSph satellites around the Milky Way3,24, and objects like these are resilient to close interaction with massive galaxies. By contrast, the very lowest mass objects in our simulation have ρ300 4.0 × 10−4 M/pc3 and Mvir 108 M, but exist only briefly as they are vulnerable to tidal disruption by large galaxies in our simulations. Together with the cutoff in the power spectrum at the Jeans scale (Supplementary Fig. 3), this leads to a marked suppression of substructure below a few times 108 M relative to the prediction of standard CDM (refs 8, 9). A quantitative evaluation of the mass function of satellite galaxies predicted by ψDM with larger simulations is thus another crucial test to be addressed.

The prominent solitonic cores uncovered in our simulations provide an opportunity to estimate the boson mass, mB, by comparison with observations, particularly for dSph galaxies where dark matter dominates. The local Fornax dSph galaxy is the best studied case, with thousands of stellar velocity measurements, allowing a detailed comparison with our soliton mass profile. We perform a Jeans analysis for the dominant intermediate metallicity stellar population, which exhibits a nearly uniform projected velocity dispersion (σ; ref. 25). We simultaneously reproduce well the radial distribution of the stars25 (Fig. 4a) and their velocity dispersion with negligible velocity anisotropy, with  eV and a core radius  kpc (Supplementary Fig. 5). The corresponding core mass M(r ≤ rc) is 9.2 × 107 M, which is hosted by a halo with virial mass 4 × 109M in the simulations. These results are similar to other estimates for Fornax5,26,27 (Fig. 4b) and consistent with other dSph galaxies derived by a variety of means4,26,28 (see Supplementary Section 3 for details).

Figure 4: Modelling the Fornax dSph galaxy with the soliton profile.
figure 4

a, Star counts of the intermediate metallicity subpopulation25 at different radial bins (symbols with 1-σ error bars) and the best-fit soliton solution (red solid line) with mB = 8.0 × 10−23 eV, rc = 0.93 kpc and σ = 11.3 km s−1, together with the 1-σ variation (red shaded). Star counts are normalized to the total number of sample stars within 2.1 kpc. Also shown are the best-fit empirical formula of Burkert35 (green dashed line) and the NFW profile (blue dot-dashed line) representing standard CDM. The scale radius of NFW is restricted to be no larger than 3.0 kpc during the fit to exclude unreasonably small concentration parameters. b, 1-σ contours of the total enclosed mass estimated from each of the three subpopulations5 (ovals), overplotted with the model curves using the same best-fit parameters adopted in a. Clearly, in both panels the soliton profile of ψDM provides an accurate fit, matched only by the empirical fitting function of the Burkert profile, whereas NFW is not favoured by the data.

For more massive galaxies, the solitons we predict are denser and more massive, scaling approximately as . So for the Milky Way, adopting a total mass of Mvir = 1012 M, we expect a soliton of Ms 2 × 109 M, with a core radius 180 pc and a potential depth corresponding to a line-of-sight velocity dispersion σ 115 km s−1 for test particles satisfying the virial condition with the soliton potential. At face value this seems consistent with the Milky Way bulge velocity dispersion, where a distinctive flat peak is observed at a level of σ 110 km s−1 within a projected radius 200 pc (refs 29, 30). Such cores clearly have implications for the creation of spheroids, acting as an essential seed for the prompt attraction of gas within a deepened potential. Indeed, bulge stars with [Fe/H] > −1.0 are firmly established as a uniformly old population that formed rapidly30,31, a conclusion that standard ΛCDM struggles to explain through extended accretion and merging30. The implications for early spheroid formation and compact nuclear objects in general can be explored self-consistently with the addition of baryons to the ψDM code, to model the interplay among stars, gas and ψDM, which will provide model rotation curves for an important test of this model.

At high redshift, the earliest galaxies formed from ψDM are delayed relative to standard CDM, limited by the small amplitude of the Jeans mass at radiation–matter equality, after which the first structures grow. This is demonstrated with a ψDM simulation of a 30 h−1 Mpc box where we adopt mB = 8.0 × 10−23 eV derived above. The first bound object collapses at z 13, with a clear solitonic core of mass 109 M and radius 300 pc, whereas under ΛCDM the first objects should form at z 50 with masses of only 104–105 M (ref. 32). The highest redshift galaxy at present at z 10.7 is multiply lensed, seeming smooth and spherical, with a stellar radius 100 pc (ref. 33), similar to local dSph galaxies. Deeper cluster lensing data from the Hubble ‘Frontier Fields’ programme will soon meaningfully explore the mass limits of galaxy formation to higher redshift, allowing us to better distinguish between particle and wavelike cold dark matter.