Axion-like ULDM: Quantum Dark Matter

• Axion-like ULDM is an ultra-light scalar dark matter model characterized by macroscopic de Broglie wavelengths and solitonic cores stabilized by quantum pressure.
• The model employs Schrödinger–Poisson dynamics to reveal unique core-halo mass relations and BEC effects, including quantized vortex formation in galactic halos.
• Experimental strategies such as strong lensing, 21-cm cosmology, and EDM measurements provide practical probes for constraining the particle mass and interactions.

Axion-like ultra-light dark matter (ULDM) refers to a class of bosonic dark matter models in which the dark matter is an extremely light scalar particle, often with a mass $m_\psi \lesssim 10^{-20}\,$eV, with “axion-like” denoting the origin of such candidates in generic light pseudo-Nambu–Goldstone bosons of spontaneously broken global symmetries. Such particles exhibit astrophysically macroscopic de Broglie wavelengths (typically kpc-scale or larger), and give rise to unique quantum phenomena in structure formation and galactic dynamics, distinct from cold dark matter (CDM). The defining property of axion-like ULDM is the appearance of kiloparsec-scale "fuzzy" features and cored density structures in gravitationally bound halos.

1. Schrödinger–Poisson Dynamics and Halo Structure

Axion-like ULDM in galactic environments is well described by the non-relativistic Schrödinger–Poisson (SP) system. The dark matter field $\psi(\mathbf{x}, t)$ evolves according to: $$i \partial_t \psi = -\frac{1}{2m} \nabla^2 \psi + m \Phi \psi$$

$\nabla^2 \Phi = 4\pi G m |\psi|^2$

where $m$ is the ULDM particle mass and $\Phi$ is the gravitational potential sourced by the ULDM density $\rho = m|\psi|^2$. The solutions to these equations reveal two universal features in equilibrium halos: an inner, self-gravitating solitonic core and an outer envelope with a Navarro–Frenk–White (NFW)-like profile arising from the excited states. The solitonic core is stabilized against gravitational collapse by quantum pressure, and all profiles admit a scaling relation $M_c r_c \approx \text{const}/m^2$, where $M_c$ and $r_c$ are the soliton mass and core radius, respectively (Zagorac et al., 2022).

The density profile of the solitonic ground state is accurately fit by

$\psi(\mathbf{x}, t)$0

with $\psi(\mathbf{x}, t)$1 the central density and $\psi(\mathbf{x}, t)$2 defined as the radius at which the density drops to half its central value. The inverse mass–radius scaling ($\psi(\mathbf{x}, t)$3) arises from the scale invariance of the SP system, so more massive solitons are more compact.

2. Core–Halo Mass Relations and Merger History Dependence

The relationship between the solitonic core mass $\psi(\mathbf{x}, t)$4 and the total host halo mass $\psi(\mathbf{x}, t)$5 in axion-like ULDM is not universal but depends on the detailed assembly history. Simulations of soliton mergers show that simultaneous mergers yield $\psi(\mathbf{x}, t)$6, while sequential (two-step) mergers give a steeper $\psi(\mathbf{x}, t)$7 scaling (Zagorac et al., 2022). The scatter in these relations can reach $\psi(\mathbf{x}, t)$8, and deviations become more pronounced as mass is "ejected" during violent relaxation or lost to simulation boundaries.

Conventional fitting of spherically averaged density profiles to the soliton shape often overestimates the core mass compared to ground-state projection by up to $\psi(\mathbf{x}, t)$9. Both approaches agree that uncertainties in the core–halo mapping are dominated by the ambiguous definition of the total halo mass, especially in the presence of ongoing mergers, accretion, and boundary losses.

In realistic cosmological halos, the core–halo relation is expected to show intrinsic scatter rather than obey a single power law, reflecting complex accretion histories and environment (Zagorac et al., 2022).

3. Quantum Scales, BEC Physics, and Vortex Phenomena

Axion-like ULDM halos possess quantum-mechanically determined characteristic scales (Lee, 2023), with the de Broglie wavelength

$$i \partial_t \psi = -\frac{1}{2m} \nabla^2 \psi + m \Phi \psi$$0

setting the characteristic core size in dwarf galaxies ($$i \partial_t \psi = -\frac{1}{2m} \nabla^2 \psi + m \Phi \psi$$1 gives $$i \partial_t \psi = -\frac{1}{2m} \nabla^2 \psi + m \Phi \psi$$2 for $$i \partial_t \psi = -\frac{1}{2m} \nabla^2 \psi + m \Phi \psi$$3 eV).

ULDM behaves as a Bose–Einstein condensate (BEC) on galactic scales, and under rotation, forms quantized vortex lattices (Zhou et al., 3 Dec 2025). The Gross–Pitaevskii–Poisson system governs the nonlinear regime, with the critical angular velocity for vortex nucleation

$$i \partial_t \psi = -\frac{1}{2m} \nabla^2 \psi + m \Phi \psi$$4

where $$i \partial_t \psi = -\frac{1}{2m} \nabla^2 \psi + m \Phi \psi$$5 is the core radius and $$i \partial_t \psi = -\frac{1}{2m} \nabla^2 \psi + m \Phi \psi$$6 the healing length, itself $$i \partial_t \psi = -\frac{1}{2m} \nabla^2 \psi + m \Phi \psi$$7–$$i \partial_t \psi = -\frac{1}{2m} \nabla^2 \psi + m \Phi \psi$$8 for typical ULDM parameters. At rotation rates $$i \partial_t \psi = -\frac{1}{2m} \nabla^2 \psi + m \Phi \psi$$9–$\nabla^2 \Phi = 4\pi G m |\psi|^2$0Myr, a lattice of vortices of $\nabla^2 \Phi = 4\pi G m |\psi|^2$1 kpc-scale spacing emerges, each carrying quantum circulation.

The underdensity columns associated with vortices alter strong lensing arcs; if observed as regular brightness anomalies separated by $\nabla^2 \Phi = 4\pi G m |\psi|^2$2, these features would constitute a unique signal of BEC-ULDM (Zhou et al., 3 Dec 2025).

4. Astrophysical and Cosmological Constraints

Observational signatures and constraints arise from multiple channels:

• Strong Lensing: The granular interference structure of ULDM halos modifies flux ratios in multiply imaged quasars. Wave interference fluctuations at the $\nabla^2 \Phi = 4\pi G m |\psi|^2$3–$\nabla^2 \Phi = 4\pi G m |\psi|^2$4 level in image magnification are generated by the density granularity, impacting mass bounds inferred from lensing (Laroche et al., 2022). Statistical analyses disfavor the lightest masses $\nabla^2 \Phi = 4\pi G m |\psi|^2$5.
• Dwarf Satellite Dynamics: The simultaneous fit of solitonic core sizes and total halo masses in Milky Way dwarf satellites is difficult in ULDM, with $\nabla^2 \Phi = 4\pi G m |\psi|^2$6 eV ruled out at $\nabla^2 \Phi = 4\pi G m |\psi|^2$7 (Safarzadeh et al., 2019).
• Galaxy Rotation Curves: Core radii and densities extracted from low-surface-brightness galaxy samples require $\nabla^2 \Phi = 4\pi G m |\psi|^2$8eV to avoid excessive "bump" features (Bar et al., 2019). Inclusion of repulsive quartic self-interactions with $\nabla^2 \Phi = 4\pi G m |\psi|^2$9 modifies the soliton–halo relation, allowing $m$0eV to be consistent with both rotation curves and core–halo scaling within GUT-allowed axion decay constants (Dave et al., 2023).

Axion-like ULDM predicts a suppression of small-scale structure below $m$1kpc. Observational non-detection of the soliton-induced rise in rotation curves of LSB and dwarf galaxies provides robust lower bounds on the particle mass. The baryonic potential must be modeled self-consistently in disk galaxies, although the core–halo relation remains robust to baryon inclusion when kinetic energy matching is used (Bar et al., 2019).

5. Nonlinear and Statistical Phenomena: Oscillons and Granularity

ULDM with nonlinear self-interactions fosters the formation of dense, long-lived localized "oscillons" via self-resonance at early times (Kawasaki et al., 2019). Typical masses $m$2 eV and moderate quartic self-interactions lead to fragmentation of the initially homogeneous field, generating oscillons of $m$3–$m$4 pc scale that persist for $m$5–$m$6 years. Such sub-kpc scale clumpiness can, in principle, affect small-scale structure, galaxy core properties, and dynamical heating processes, and necessitates the reevaluation of Lyman-$m$7 and direct detection limits under non-smooth ULDM distributions.

Beyond deterministic structure, even in free theory, wave interference effects render the halo density field "granular" on the de Broglie scale, perturbing both image fluxes in lensing and fundamentally violating the collisionless-fluid description assumed in CDM (Laroche et al., 2022).

6. Detection Strategies and Experimental Probes

The unique features of axion-like ULDM motivate diverse detection strategies:

• Cosmological Birefringence: Axion-photon coupling causes oscillating birefringence signatures in laser interferometer arms. LISA-like interferometers, modified to be polarization-sensitive, can probe couplings $m$8–$m$9 at masses $\Phi$0–$\Phi$1 eV using Sagnac time-delay interferometry, opening new reach below established helioscope and astrophysical limits (Yao et al., 2024).
• 21-cm Cosmology: ULDM-induced baryon cooling and ALP–photon resonant conversion can both enhance the 21-cm absorption trough during cosmic dawn (Das, 2024). The interplay of cooling (via BEC-mediated energy transfer) and heating (via magnetic-field-driven ALP–photon conversion) provides a parameter regime in which observed anomalies (such as the EDGES signal) can be reproduced for $\Phi$2 eV, $\Phi$3–$\Phi$4 GeV$\Phi$5.
• Electric Dipole Moments: For ultra-light axions ($\Phi$6 eV) comprising the entire DM abundance, loop-induced EDMs (from CP-odd ALP couplings) put stringent bounds on products of couplings: for example, $\Phi$7 for $\Phi$8 eV, with EDM sensitivity surpassing previous bounds by $\Phi$9–$\rho = m|\psi|^2$0 orders of magnitude (Evans et al., 16 Sep 2025).

7. Open Issues and Future Directions

The axion-like ULDM scenario remains subject to both theoretical and empirical uncertainties. The core–halo relation displays significant scatter tied to hierarchical assembly and environment, complicating inferences from rotation curves and lensing (Zagorac et al., 2022). The coexistence of BEC, vortex formation, and possible oscillon substructure demands high-resolution, multi-component simulations that incorporate baryons and allow for realistic cosmological context. Dwarf satellite populations currently challenge pure minimal ULDM, but inclusion of self-interactions and careful halo assignment may ameliorate the tension (Dave et al., 2023).

Planned and next-generation observational campaigns—including highly resolved strong-lens imaging, wide-field time-delay cosmography, and networked precision interferometers—have the potential to decisively test axion-like ULDM in the $\rho = m|\psi|^2$1–$\rho = m|\psi|^2$2 eV mass window. The model’s distinctive predictions—soliton cores, quantum granularity, vortex lensing signatures, oscillonic substructure, and cosmological birefringence—render it falsifiable with upcoming data, provided theoretical models continue to bridge the gap between simulated and observed systems (Zhou et al., 3 Dec 2025, Yao et al., 2024, Das, 2024, Zagorac et al., 2022, Laroche et al., 2022, Safarzadeh et al., 2019, Dave et al., 2023, Bar et al., 2019, Kawasaki et al., 2019, Lee, 2023, Kendall et al., 2019).

---

References:

(Laroche et al., 2022, Zagorac et al., 2022, Zhou et al., 3 Dec 2025, Yao et al., 2024, Bar et al., 2019, Kawasaki et al., 2019, Blum et al., 2024, Dave et al., 2023, Das, 2024, Lee, 2023, Safarzadeh et al., 2019, Evans et al., 16 Sep 2025, Kendall et al., 2019)