Self-Interacting Fuzzy Dark Matter

• SIFDM is an extension of fuzzy dark matter that incorporates ultra-light bosonic fields with quartic self-interactions to modify solitonic core properties and core-halo scaling.
• The theoretical framework employs a coupled Gross–Pitaevskii–Poisson system to model both repulsive and attractive self-interactions and their impact on quantum turbulence and collapse.
• Astrophysical implications include altered rotation curves, modified soliton cores, and potential seeding of supermassive black holes, providing testable predictions against cosmological data.

Self-Interacting Fuzzy Dark Matter (SIFDM) is an extension of the standard Fuzzy Dark Matter (FDM) paradigm in which ultra-light bosonic particles not only obey quantum wave dynamics but also interact via a (typically quartic) self-interaction. This framework unifies the phenomenology of non-interacting FDM, standard Cold Dark Matter (CDM), and regimes dominated by scalar field self-interaction. SIFDM models are motivated both from fundamental field theory (e.g., axion-like particles with symmetry-breaking scales) and from astrophysical needs to explain galactic core structure, suppress small-scale cosmological power, and potentially seed supermassive black holes. The SIFDM scenario incorporates both repulsive and attractive couplings, modifies the solitonic core properties, alters the core-halo connection, introduces new dynamical phenomena such as quantum turbulence and collapse transitions, and yields testable deviations from vanilla FDM observable in precision galactic and cosmological data.

1. Theoretical Formalism and Governing Equations

The fundamental degrees of freedom in SIFDM are nonrelativistic bosonic fields, typically modeled as a complex scalar $\psi(\mathbf{x},t)$ with mass $m$ and contact quartic self-coupling $\lambda$. The SIFDM evolution equations, valid for both repulsive ($\lambda > 0$) and attractive ($\lambda < 0$) interactions, are generally formulated as a coupled Gross–Pitaevskii–Poisson (GPP) system:

$$i\hbar\,\frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2 \psi + m \Phi \psi + \lambda |\psi|^2 \psi$$

$$\nabla^2 \Phi = 4 \pi G m \left(|\psi|^2 - \langle |\psi|^2 \rangle \right)$$

Here, $\Phi(\mathbf{x},t)$ is the gravitational potential and $\langle|\psi|^2\rangle$ is the mean density (subtracted to enforce periodicity or cosmological background). The quartic self-interaction term may be mapped to a two-body s-wave scattering length $a_s$ via $m$0. For ultra-light axion models, $m$1 is dictated by the symmetry-breaking scale $m$2 according to $m$3 (Painter et al., 2024, Mocz et al., 2023).

For mixed regimes (condensate + particle excitations), first-principles derivations employing the Schwinger-Keldysh formalism yield a modified Gross-Pitaevskii equation for the condensate, a kinetic (Boltzmann-type) equation for higher-momentum quasi-particles, and a Poisson equation for gravity, with interaction and collision terms representing condensate-particle (Beliaev-type) and particle-particle (2↔2) scattering processes up to $m$4 (Proukakis et al., 2023).

2. Soliton Core Structure and Scaling Relations

The internal structure of SIFDM halos is governed by the competition between quantum pressure (arising from the kinetic term), gravity, and bosonic self-interactions. Solitonic (groundstate) solutions exhibit qualitatively different properties depending on the sign and strength of the interaction.

Key limiting regimes:

• Pure FDM (λ = 0): Soliton core radius scales as $m$5 with $m$6 the soliton mass, central density $m$7 (Indjin et al., 30 Jun 2025, Indjin et al., 2023).
• Repulsive SIFDM (λ > 0): At strong coupling (Thomas–Fermi limit), $m$8 becomes mass-independent for large $m$9, while $\lambda$0 (Indjin et al., 30 Jun 2025, Indjin et al., 2023, López-Sánchez et al., 7 Dec 2025).
• Attractive SIFDM (λ < 0): There exists a maximum allowable mass $\lambda$1 beyond which solitons are unstable to collapse, with high-density post-collapse objects stabilized by higher-order interactions or general relativistic effects (Painter et al., 2024, Mocz et al., 2023, López-Sánchez et al., 7 Dec 2025).

Generalized virial relations:

For any $\lambda$2 (dimensionless interaction strength, with $\lambda$3 set by equating self-interaction and quantum-kinetic energies), the soliton structure can be analytically solved via parameterized ansätze and shape parameters (Indjin et al., 2023, Indjin et al., 30 Jun 2025). Simulations confirm the analytic scaling relations across parameter space (López-Sánchez et al., 7 Dec 2025).

3. Core–Halo Relations and Astrophysical Consequences

The inclusion of self-interactions fundamentally modifies the mass–radius relation of solitonic cores and the core–halo mass scaling:

• Repulsive SI: Cores are more massive and extended, with lower central densities than pure FDM, relaxing constraints from galaxy rotation curves and matching observed core radii in dwarfs and LSB galaxies (Indjin et al., 7 Feb 2025, López-Sánchez et al., 7 Dec 2025).
• Attractive SI: Enhances central densities and can trigger fast collapse into compact objects ("bosenova" events), providing a channel for non-baryonic seeding of supermassive black holes, especially relevant for massive halos (Painter et al., 2024, Mocz et al., 2023, López-Sánchez et al., 7 Dec 2025).

Core–halo scaling relations take the form $\lambda$4 with $\lambda$5 weakly dependent on self-interaction and $\lambda$6 a dimensionless energy invariant (López-Sánchez et al., 7 Dec 2025). Dynamical simulations using observed galactic rotation curves (e.g., SPARC database) provide joint constraints on $\lambda$7; best-fit regions are found near $\lambda$8 eV, $\lambda$9 J m$\lambda > 0$0/kg (Indjin et al., 7 Feb 2025).

4. Dynamical Phenomena: Oscillations, Vortices, and Collapse

SIFDM halos exhibit distinctive dynamical structures and processes:

• Oscillation Spectra: The central soliton supports breathing-mode oscillations with frequencies governed by central density and interaction strength—repulsive SI lowers the oscillation frequency (Indjin et al., 30 Jun 2025, Indjin et al., 2023).
• Vortices and Turbulence: Quantum turbulence in the outer halo manifests as persistent vortex networks observable via coherence measures and incompressible kinetic energy spectra; these networks are robust under moderate self-interaction (Indjin et al., 30 Jun 2025).
• Collapse in Attractive SIFDM: Cores exceeding $\lambda > 0$1 undergo catastrophic collapse on $\lambda > 0$2 Myr timescales, producing unresolved compact remnants accompanied by energy emission, with implications for early SMBH formation (Painter et al., 2024, Mocz et al., 2023, López-Sánchez et al., 7 Dec 2025).

5. Cosmological Structure Formation and Wave Interference Effects

In cosmology, SIFDM modifies both linear and nonlinear structure formation:

• Linear Regime: Repulsive SI shifts the core scale (via the TF radius) and damps granular power; attractive SI enhances small-scale structure by partially restoring power suppressed in pure FDM (Mocz et al., 2023).
• Nonlinear Dynamics: Wave interference strongly regulates the growth of density perturbations on timescales $\lambda > 0$3 (de Broglie crossing) rather than fluid-like self-interaction timescales, confining SI-induced deviations to $\lambda > 0$4 in most halos (Capanelli et al., 27 Mar 2025). This limits the astrophysical impact of self-interaction except in environments with exceptionally high densities and small velocity dispersions—e.g., ultrafaint dwarfs with compact TF cores.

6. Observational Signatures and Constraints

SIFDM models predict several direct signatures:

• Core properties: Modified soliton profiles (super-Gaussian slopes, universal inner rotation curves), increased core sizes for repulsive SI, and critical collapse events for attractive SI (Indjin et al., 30 Jun 2025, Indjin et al., 7 Feb 2025, Painter et al., 2024).
• Rotation curves and lensing: Fitted core–halo models yield rotation curves and projected mass profiles differing from both CDM and pure FDM, allowing for joint constraints on $\lambda > 0$5 with current and forthcoming high-resolution data (Indjin et al., 7 Feb 2025, López-Sánchez et al., 7 Dec 2025).
• Lyman-α Forest and Small-scale Power: Attractive SI can partially restore the small-scale power deficit of FDM, potentially alleviating the FDM “Catch-22” tension between Lyman-$\lambda > 0$6 constraints and core radii in galaxies (Mocz et al., 2023).
• Black Hole Demographics: The presence or absence of SMBHs in dwarf galaxies constrains attractive SI parameters via the collapse threshold mass and predicted formation timescales (López-Sánchez et al., 7 Dec 2025).
• Vortex dynamics and granulation: Sub-kiloparsec lensing perturbations and dynamical heating of stellar populations in the presence of quantum turbulence or TF-dominated cores offer further diagnostic channels (Indjin et al., 30 Jun 2025, Capanelli et al., 27 Mar 2025).

Cosmological (e.g., Lyman-α) and galactic (e.g., SPARC, UFD kinematics) observations jointly constrain the allowable parameter space, generally requiring $\lambda > 0$7 cm for $\lambda > 0$8 eV to match core properties without violating bounds from the absence of excessive SMBHs or overly large cores (López-Sánchez et al., 7 Dec 2025, Painter et al., 2024, Indjin et al., 7 Feb 2025).

7. Unified Frameworks and Future Directions

Recent analytical work provides a unified description interpolating between cold, fuzzy, and self-interacting dark matter by coupling condensate, kinetic, and gravitational dynamics within a controlled approximation (Popov, gradient expansion), permitting coexistence of condensed and particle phases, as well as the inclusion of self-interaction and collisional effects (Proukakis et al., 2023). Direct dynamical reconstructions from realistic initial conditions demonstrate that the observed galactic halo population is consistent with SIFDM for a single parameter set $\lambda > 0$9 (Indjin et al., 7 Feb 2025). Semi-analytical and numerical studies emphasize that static and dynamic soliton features (e.g., size, density, breathing frequency) are degenerate in pure FDM but jointly constrain $\lambda < 0$0 uniquely in SIFDM, motivating multi-messenger and multi-scale observational campaigns (Indjin et al., 2023, Indjin et al., 30 Jun 2025).

Future developments include full general relativistic simulations of collapse, improved embedding of SIFDM physics into N-body/hydrodynamical cosmological codes, and further tests against lensing and kinematic surveys targeting both core and halo phenomenology (Painter et al., 2024, Mocz et al., 2023).