Fuzzy Dark Matter Dynamics

• Fuzzy Dark Matter is a dark matter paradigm featuring ultralight bosonic particles with kiloparsec-scale de Broglie wavelengths that produce persistent quantum fluctuations.
• Its wave-driven dynamics lead to measurable phenomena such as inspiral stalling, modified dynamical friction, and stochastic heating in galactic systems.
• The framework employs Schrödinger–Poisson equations and kinetic models to translate quantum interference effects into observable astrophysical predictions.

Fuzzy Dark Matter (FDM) is a dark matter paradigm in which the dark sector is composed of ultralight bosonic particles, typically with masses $m_b \sim 10^{-22}\,\mathrm{eV}$, leading to macroscopic de Broglie wavelengths on the order of kiloparsecs in galactic halos. The wave nature of the FDM field results in persistent, stochastic density and potential fluctuations that do not damp over cosmological timescales. These non-classical aspects produce distinctive relaxation, dynamical friction, and heating phenomena in galactic environments that differ sharply from conventional cold dark matter (CDM) models. The relaxation processes and their consequences provide both a direct test of FDM on galactic scales and a mechanism to use astrophysical observations to constrain the properties of the dark matter particle (Bar-Or et al., 2018).

1. Wave Dynamics and Granular Fluctuations

In FDM, the gravitational potential and density field are determined by a macroscopic wavefunction obeying the Schrödinger–Poisson equations rather than by individual, collisionless particles as in CDM. For an FDM boson of mass $m_b$, each mode with velocity dispersion $\sigma$ features a de Broglie wavelength

$\lambda = h/(m_b \sigma)$

which, for $m_b \approx 10^{-22}\,\mathrm{eV}$ and $\sigma \sim 200$ km/s, gives $\lambda \approx 1$ kpc. On scales below $\lambda$, quantum interference between many plane-wave modes leads to persistent, granular density fluctuations—a phenomenon absent in CDM. These “granules” can be regarded as effective quasiparticles with a characteristic mass determined by formulation: $$m_{\mathrm{eff}} = \frac{\pi^{3/2} \hbar^3}{m_b^3 \sigma^3}$$ For typical galactic parameters, $$m_{\mathrm{eff}} \simeq 10^7 M_\odot (1\,\mathrm{kpc}/r)^2 (10^{-22}\,\mathrm{eV}/m_b)^3$$.

Granular fluctuations in FDM are fundamentally non-damping; they source stochastic gravitational fields that impact the orbits of stars, black holes, and any other massive tracers embedded within FDM halos (Bar-Or et al., 2018).

2. Analogy to Classical Two-body Relaxation

Despite its quantum origin, FDM halo relaxation can be cast in close analogy with classical two-body (collisional) relaxation in gravitational $N$-body systems. The stochastic FDM-induced force field leads to diffusion in orbital energy and angular momentum that can be quantified using Fokker–Planck equations with modified coefficients:

• In the standard $N$-body case, the relaxation process is determined by distant encounters with point masses of mass $m$ and a minimum impact parameter set by the system's resolution scale.
• In FDM, the stochastic "kick" is mediated by granules of mass $m_{\mathrm{eff}}$, and the minimum effective impact parameter is set by $\lambda/2$.

The velocity diffusion coefficient in FDM halos has the form

$$D[\Delta v] \sim 4\pi G^2 m_{\mathrm{eff}} \ln\Lambda_{\mathrm{FDM}}(\ldots)$$

where $\ln\Lambda_{\mathrm{FDM}}$ is a Coulomb logarithm with upper cutoff set by the system scale and lower cutoff at $\lambda/2$ (Bar-Or et al., 2018). The relaxation time, dominated by these persistent stochastic fluctuations, is

$$T_{\mathrm{heat}} = \frac{3 m_b^3 \sigma^6}{16\pi^2 G^2 \hbar^3 \ln\Lambda_{\mathrm{FDM}}}$$

The analogy is formal: the underlying process is not that of discrete encounters, but emerges from the persistent, macroscopic interference.

3. Dynamical Friction, Mass Segregation, and Inspiral Stalling

FDM fluctuations not only induce stochastic diffusion but also modify dynamical friction. The drag force on a massive particle ($m_t$) is analogous to the classical Chandrasekhar formula but altered as follows:

• For $m_t \gg m_{\mathrm{eff}}$, classical dynamical friction dominates, leading to inspiral.
• As the inspiralling object approaches $m_t \sim m_{\mathrm{eff}}$, the stochastic, diffusive kicks from FDM granules grow in significance and counteract frictional drag.
• When $m_t \sim m_{\mathrm{eff}}$, the balance leads to “stalling” at a characteristic radius: $$r_{\mathrm{stall}} \sim \left( \frac{\hbar^3}{G m_t m_b^3} \right)^{1/2}$$ For $m_t \sim 10^7 M_\odot$ and $m_b \sim 10^{-22}$ eV, this radius is a few hundred parsecs.

This mechanism has major implications:

• The inspiral of supermassive black holes or globular clusters is expected to stall at $r \sim r_{\mathrm{stall}}$, potentially inhibiting SMBH coalescence or the infall of massive clusters over a Hubble time.
• Mass segregation of dense objects (e.g., binaries, remnant populations) in central galactic and nuclear regions is suppressed when their mass is not much greater than $m_{\mathrm{eff}}$ (Bar-Or et al., 2018).

4. Stochastic Heating and Expansion of Stellar Systems

The same stochastic FDM potential gives rise to secular heating of embedded stellar populations. If the relaxation time $T_{\mathrm{heat}}$ is less than the Hubble time, the cumulative random energy kicks “puff up” the stellar velocity dispersion and expand the spatial distribution. Specifically,

$$T_{\mathrm{heat}} = \frac{3 \sigma^3}{16\sqrt{\pi} G^2 m_{\mathrm{eff}} \ln\Lambda_{\mathrm{FDM}}}$$

where $\sigma$ is the local velocity dispersion. For $m_b \sim 10^{-22}$ eV and typical inner galaxy densities, this time can be comparable to or shorter than the age of the galaxy, leading to observable consequences:

• Stellar density cores or depleted inner regions in systems where CDM would predict cuspy central concentrations.
• Expansion (increase of half-light radius) of compact nuclear clusters or galactic spheroids over $\sim 10$ Gyr.

These predictions are testable against the demographics of nuclear star clusters and the phase-space density of stars in local galaxies (Bar-Or et al., 2018).

5. Sensitivity to the FDM Particle Mass and Observational Implications

Both the effective granule mass $m_{\mathrm{eff}}$ and the stochastic heating timescale scale strongly with $m_b$:

• $m_{\mathrm{eff}} \propto m_b^{-3}$
• $\lambda \propto m_b^{-1}$
• $T_{\mathrm{heat}} \propto m_b^3$

Thus, even modest changes in $m_b$ have large effects on predicted dynamical signatures. By comparing the expected location of inspiral stalling, the degree of central heating, or the lack of massive objects close to galaxy centers with high-resolution observations, stringent constraints on $m_b$ can be set. Specifically:

• If massive objects are found to spiral into the centers of galaxies as expected in CDM, then $m_b$ must be larger than $\sim 10^{-22}$ eV.
• If low-density cores and expanded stellar systems are seen where CDM predicts concentrations, it can favor a specific FDM mass range.

Current and future surveys targeting the dynamics and spatial distributions of SMBHs, globular clusters, and inner stellar populations in faint and low-surface-brightness galaxies will provide crucial empirical tests of these predictions (Bar-Or et al., 2018).

6. Theoretical and Modeling Implications

Treating the FDM field as a collection of effective quasiparticles provides a tractable framework for importing kinetic theory tools, such as the Fokker–Planck approach, into the regime of macroscopic, fluctuating wave-like dark matter. The correspondence is, however, only formal; FDM fluctuations are non-local and “coherent” on scales of $\lambda$, unlike the uncorrelated classical particles in CDM. The choice of granule mass ($m_{\mathrm{eff}}$), the impact parameter ($\sim \lambda/2$), and the form of the Coulomb logarithm are all directly tied to the wave properties of the dark matter, making the structure of FDM halos a sensitive probe of both microphysical parameters and macroscopic dynamics.

A precise translation between the predicted dynamical effects and their simulation or Fokker–Planck modeling requires careful treatment of:

• The transition between grainy FDM-induced stochasticity and classical two-body encounters.
• The role of baryonic tracers versus dark halo self-relaxation.
• The impact of inhomogeneity, anisotropy, and non-static backgrounds.

7. Summary Table: Key FDM Relaxation Quantities

Quantity	Formula	Comments
Effective granule mass	$$m_{\mathrm{eff}} = \dfrac{\pi^{3/2} \hbar^3}{m_b^3 \sigma^3}$$	Strong $m_b^{-3}$ scaling
de Broglie wavelength	$\lambda = h/(m_b \sigma)$	Sets scale for granularity
Inspiral stalling radius	$$r_{\mathrm{stall}} \sim [\hbar^3/(G m_t m_b^3)]^{1/2}$$	For $m_t \sim m_{\mathrm{eff}}$
Heating timescale	$$T_{\mathrm{heat}} = \dfrac{3 m_b^3 \sigma^6}{16 \pi^2 G^2 \hbar^3 \ln \Lambda_{\mathrm{FDM}}}$$	Cumulative effect on stellar systems

All expressions above are derived in (Bar-Or et al., 2018), with numerical prefactors and dependencies subject to the choice of halo structure and local parameters.

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In summary, the relaxation dynamics of FDM halos—driven by persistent, stochastic, macroscopic density fluctuations sourced by interference among ultralight bosonic modes—predict novel and quantifiable deviations from CDM expectations. The effective theory of "quasiparticle" granules enables the adaptation of kinetic theory to wave dark matter and yields explicit predictions for inspiral stalling of massive objects, stochastic heating of inner stellar systems, and the scale dependence of these effects on the underlying bosonic mass. Comparison with advanced astrometric and dynamical measurements thus enables the use of galactic structural data as a sensitive probe of the fundamental properties of the dark sector.