Fuzzy Dark Matter Dynamical Friction

• FDM dynamical friction is the gravitational drag arising when a massive perturber moves through an ultralight-boson dark matter field described by the Schrödinger–Poisson system.
• Linear-response theory shows that quantum wave effects and de Broglie-scale coherence modify the classical Chandrasekhar drag, altering wake morphology and torque.
• Both simulations and semi-analytic models reveal that interference granules, solitonic cores, and granule-induced diffusion can stall inspiral and reshape subhalo dynamics.

Searching arXiv for recent and foundational papers on fuzzy dark matter dynamical friction. Found relevant papers spanning semi-analytic subhalo evolution, linear-response theory, Schrödinger–Poisson simulations, globular cluster inspiral, SMBH/soliton dynamics, and wake observables. Fuzzy Dark Matter (FDM) dynamical friction is the gravitational drag, stochastic heating, and associated wake dynamics that arise when a massive perturber moves through an ultralight-boson dark-matter medium described by the Schrödinger–Poisson system rather than by a purely collisionless particle bath. In this setting, the classical Chandrasekhar picture remains a useful baseline, but it is modified by de Broglie-scale coherence, quantum-pressure regularization, interference granules, and, in many halos, solitonic cores. As a result, the effective drag can be weakened, time-dependent, morphology-dependent, or balanced by diffusion; in some regimes it remains close to the cold-dark-matter expectation, while in others it stalls inspiral or is subdominant to other FDM-specific dissipation channels (Lancaster et al., 2019).

1. Classical baseline and the FDM medium

The classical reference problem is a perturber of mass $M$ moving with speed $v$ through a homogeneous collisionless background of density $\rho$ and velocity dispersion $\sigma$. In that limit, Chandrasekhar’s formula gives

$$F_{\rm classical} = -\,4\pi\,G^2\,M^2\,\frac{\rho}{v^2}\,\ln\Lambda \left[ \operatorname{erf}(X)-\frac{2X}{\sqrt{\pi}}e^{-X^2} \right], \qquad X\equiv \frac{v}{\sqrt{2}\sigma}.$$

Its structure isolates two elements: the $M^2\rho/v^2$ scaling and the Coulomb logarithm $\ln\Lambda$, which summarizes the impact-parameter range relevant to wake formation (Lancaster et al., 2019).

FDM replaces the collisionless medium by an ultralight bosonic field obeying the Schrödinger–Poisson equations,

$$i\hbar\,\partial_t\psi = -\frac{\hbar^2}{2m_a}\nabla^2\psi + m_a\Phi\,\psi, \qquad \nabla^2\Phi = 4\pi G\,\rho, \qquad \rho = m_a|\psi|^2,$$

or, in Madelung form, by an irrotational fluid with a quantum-pressure term. The characteristic coherence scale is the de Broglie wavelength,

$\lambda_{\rm dB}=\frac{h}{m_a v},$

which can be astrophysically large and therefore directly regulate the wake on galactic scales (Foote et al., 2023).

This wave character changes both the morphology and the interpretation of dynamical friction. Rather than a purely downstream overdensity generated by incoherent particle deflections, the perturber excites a dispersive density response with interference, finite coherence length, and, depending on regime, persistent granular fluctuations. Several studies therefore distinguish between the classical drag term itself and the broader orbital evolution generated by drag, stochastic heating, and collective FDM modes (Bar-Or et al., 2018).

2. Linear-response theory and wave-regulated drag

A central result of linear-response theory is that FDM modifies the dynamical-friction kernel by suppressing short-scale response below the de Broglie scale. For a point mass in a uniform background, the steady-state drag can still be written as

$$F_{\rm FDM} = -\,4\pi G^2 M^2 \rho\,\frac{C_{\rm rel}}{v_{\rm rel}^2},$$

but the dimensionless coefficient $v$0 is no longer a Coulomb logarithm derived from classical two-body scattering. In the linear perturbative regime it is controlled by the wave response and, for a point mass, is approximated by

$v$1

with $v$2 measured in units of $v$3. In this formulation, FDM effectively imposes a minimum scale of order $v$4 even for a point perturber; extended satellites introduce an additional form-factor suppression through their physical size (Lancaster et al., 2019).

For circular orbits, the linearized hydrodynamic form yields a forced quantum-wave equation for the density contrast,

$v$5

The corresponding drag separates naturally into radial and tangential components through a complex response function $v$6,

$v$7

with $v$8. In this formulation, the short-distance Coulomb divergence of gaseous or collisionless treatments is absent, but an infrared divergence appears in the steady-state radial component. The finite-time problem regularizes this by allowing the wake to diffuse only out to a finite outer boundary; after switch-on, both $v$9 and $\rho$0 oscillate about the steady-state values with a decaying envelope, so strict steady state is never reached (2207.13740).

Velocity dispersion further smooths the wake. In the wave formulation, this is naturally parameterized by the ratio $\rho$1. When $\rho$2 does not significantly exceed $\rho$3, the analytic circular-orbit solution remains accurate, whereas $\rho$4 substantially smooths the response and can strongly suppress the torque (2207.13740).

3. Granules, diffusion, and stall criteria

A defining feature of FDM halos is the presence of persistent density fluctuations generated by interference of the wave field. These fluctuations can be treated as quasiparticles with an effective mass

$\rho$5

or, equivalently in the relaxation treatment, as fluctuations with effective mass proportional to $\rho$6. This mapping allows relaxation and dynamical-friction calculations to be written in forms analogous to classical two-body theory, but with a wave-imposed Coulomb logarithm and an effective bath set by the granules rather than by individual particles (Bar-Or et al., 2018).

In this picture, orbital evolution is determined by competition between a mass-proportional drag and stochastic heating. A compact statement of the balance condition is

$\rho$7

which marks the onset of inspiral stalling: the kinetic energy injected by FDM granules balances the energy lost to friction. In an isothermal halo, this produces the explicit estimate

$\rho$8

so lighter bosons and lower perturber masses move the stall radius outward (Bar-Or et al., 2018).

Direct simulations of nuclear objects in an isolated FDM halo sharpen this picture inside and around the soliton. In a $\rho$9 halo with $\sigma$0, objects with mass $\sigma$1 of the soliton mass are expelled from the soliton in $\sigma$2 and then continue outward diffusion, whereas objects with mass $\sigma$3 remain largely confined because dynamical friction dominates. The same simulations identify soliton oscillations with periods $\sigma$4–$\sigma$5 and center-of-density excursions up to $\sigma$6, establishing that “dynamical friction in FDM” cannot be separated from time-dependent potential fluctuations generated by the soliton and the surrounding granules (Chowdhury et al., 2021).

4. Geometry, circular motion, and non-linear wake dynamics

The morphology of the perturber matters more strongly in FDM than in the classical spherical treatment. Simulations of satellites described by axisymmetric logarithmic potentials show that the same satellite may experience a drag differing by a factor of $\sigma$7 depending on its ellipticity and the direction of motion. In these calculations, wakes differ qualitatively from spherical cases, with upstream oscillatory features and, at low quantum Mach number, transient sign reversals in the drag coefficient. For a $\sigma$8 satellite traversing at $\sigma$9 a halo of mean density $$F_{\rm classical} = -\,4\pi\,G^2\,M^2\,\frac{\rho}{v^2}\,\ln\Lambda \left[ \operatorname{erf}(X)-\frac{2X}{\sqrt{\pi}}e^{-X^2} \right], \qquad X\equiv \frac{v}{\sqrt{2}\sigma}.$$0, the resulting dynamical-friction timescale is close to Hubble time (Vitsos et al., 2022).

Circular motion introduces additional structure. In the analytic circular-orbit treatment, the wake is diffusive rather than shock-like, and the tangential torque can change sign in binaries. The paper explicitly notes that compact-binary inspiral may stall because the DF torque about the binary center of mass sometimes flips sign to become a thrust rather than a drag. This is a specific consequence of the FDM wave response and has no Chandrasekhar analogue (2207.13740).

Fully self-consistent Schrödinger–Poisson simulations add genuinely non-linear behavior. In a uniform ULDM medium, the wake behind a large moving point mass can become self-gravitating and collapse, dramatically increasing the drag force and rapidly halting the perturber, although that regime is judged unlikely to be astrophysically common. Inside a ULDM soliton, a moving supermassive black hole excites coherent “breathing modes” of the soliton, producing “stone skipping” trajectories and stochastic motion near the center rather than monotonic sinking. For the fiducial soliton considered there, the characteristic decay-time estimate at the half-mass radius is

$$F_{\rm classical} = -\,4\pi\,G^2\,M^2\,\frac{\rho}{v^2}\,\ln\Lambda \left[ \operatorname{erf}(X)-\frac{2X}{\sqrt{\pi}}e^{-X^2} \right], \qquad X\equiv \frac{v}{\sqrt{2}\sigma}.$$1

but the simulations show order-unity deviations because the soliton response is non-stationary (Wang et al., 2021).

Another non-linear regime appears in head-on collisions of compact FDM subhalos. There, the dominant dissipative channel at low encounter speed is gravitational cooling rather than ordinary wake drag. For $$F_{\rm classical} = -\,4\pi\,G^2\,M^2\,\frac{\rho}{v^2}\,\ln\Lambda \left[ \operatorname{erf}(X)-\frac{2X}{\sqrt{\pi}}e^{-X^2} \right], \qquad X\equiv \frac{v}{\sqrt{2}\sigma}.$$2, the measured velocity loss is $$F_{\rm classical} = -\,4\pi\,G^2\,M^2\,\frac{\rho}{v^2}\,\ln\Lambda \left[ \operatorname{erf}(X)-\frac{2X}{\sqrt{\pi}}e^{-X^2} \right], \qquad X\equiv \frac{v}{\sqrt{2}\sigma}.$$3 in FDM versus $$F_{\rm classical} = -\,4\pi\,G^2\,M^2\,\frac{\rho}{v^2}\,\ln\Lambda \left[ \operatorname{erf}(X)-\frac{2X}{\sqrt{\pi}}e^{-X^2} \right], \qquad X\equiv \frac{v}{\sqrt{2}\sigma}.$$4 in CDM, while at $$F_{\rm classical} = -\,4\pi\,G^2\,M^2\,\frac{\rho}{v^2}\,\ln\Lambda \left[ \operatorname{erf}(X)-\frac{2X}{\sqrt{\pi}}e^{-X^2} \right], \qquad X\equiv \frac{v}{\sqrt{2}\sigma}.$$5 both models give $$F_{\rm classical} = -\,4\pi\,G^2\,M^2\,\frac{\rho}{v^2}\,\ln\Lambda \left[ \operatorname{erf}(X)-\frac{2X}{\sqrt{\pi}}e^{-X^2} \right], \qquad X\equiv \frac{v}{\sqrt{2}\sigma}.$$6. The fitted scaling

$$F_{\rm classical} = -\,4\pi\,G^2\,M^2\,\frac{\rho}{v^2}\,\ln\Lambda \left[ \operatorname{erf}(X)-\frac{2X}{\sqrt{\pi}}e^{-X^2} \right], \qquad X\equiv \frac{v}{\sqrt{2}\sigma}.$$7

identifies gravitational cooling, not wave-mediated dynamical friction, as the dominant source of the observed FDM deceleration in that setup (Koo, 1 Jul 2025).

5. Subhalos, merger trees, and semi-analytic prescriptions

In semi-analytic galaxy-formation calculations, “FDM dynamical friction” often appears in a more indirect form. A representative implementation constructs merger trees with an FDM-specific halo mass function based on a mass-dependent excursion-set barrier,

$$F_{\rm classical} = -\,4\pi\,G^2\,M^2\,\frac{\rho}{v^2}\,\ln\Lambda \left[ \operatorname{erf}(X)-\frac{2X}{\sqrt{\pi}}e^{-X^2} \right], \qquad X\equiv \frac{v}{\sqrt{2}\sigma}.$$8

so that suppressed small-scale power and scale-dependent growth first reduce the abundance of low-mass progenitors. In this framework, dynamical friction then acts on the surviving satellite population, which is already biased toward larger masses (Du et al., 2016).

The actual orbital decay prescription in GALACTICUS remains the standard Chandrasekhar acceleration,

$$F_{\rm classical} = -\,4\pi\,G^2\,M^2\,\frac{\rho}{v^2}\,\ln\Lambda \left[ \operatorname{erf}(X)-\frac{2X}{\sqrt{\pi}}e^{-X^2} \right], \qquad X\equiv \frac{v}{\sqrt{2}\sigma}.$$9

with $M^2\rho/v^2$0 in the orbiting runs used for the substructure statistics. No FDM-specific Coulomb logarithm or wave-wake correction is introduced; the parameters are CDM-calibrated (Du et al., 2016).

Within that setup, the main suppression of the subhalo mass function does not come from dynamical friction alone. In the CDM baseline, the “simple” merging-time model and the orbiting model with dynamical friction only give broadly consistent subhalo mass functions, whereas adding tidal stripping significantly reduces and shifts the subhalo mass function to lower masses. In mixed CDM+FDM models, the low-mass substructure suppression is stronger than in CDM, but the paper explicitly finds that tidal stripping dominates the reduction when combined with the FDM-induced cutoff in the halo mass function (Du et al., 2016).

Pure FDM introduces an additional modification through solitonic cores. In the satellite model, stripping is halted when the bound mass approaches $M^2\rho/v^2$1, and core–core contact defines rapid merging. This generates a bump in the $M^2\rho/v^2$2 subhalo mass function peaking around $M^2\rho/v^2$3–$M^2\rho/v^2$4, interpreted as long-lived naked-core satellites that survive stripping and continue inward migration until the core-merger threshold is met. In this semi-analytic context, therefore, the most important FDM effects on substructure arise indirectly through progenitor suppression and altered stripping/merging rules, not through an intrinsically modified dynamical-friction law (Du et al., 2016).

6. Globular clusters, galaxy wakes, and observational leverage

Globular-cluster systems in dwarfs provide a particularly direct application. A recent implementation in galpy introduces an FDM dynamical-friction force with coefficient $M^2\rho/v^2$5, recomputed along the orbit and constrained never to exceed the classical CDM value. In this treatment, FDM density granules reduce or suppress classical drag, and inspiral stalls when $M^2\rho/v^2$6. For host halos $M^2\rho/v^2$7–$M^2\rho/v^2$8 and $M^2\rho/v^2$9, three regimes are identified: Zone 1 with $\ln\Lambda$0 has inefficient DF or stalling, Zone 2 with $\ln\Lambda$1 has reduced DF, and Zone 3 with $\ln\Lambda$2 approaches the classical limit (Szpilfidel et al., 1 Oct 2025).

The Fornax timing problem is one concrete benchmark. In the galpy implementation with $\ln\Lambda$3, a realistic FDM core plus FDM dynamical friction makes GC3 stall around $\ln\Lambda$4 even after $\ln\Lambda$5; with an NFW halo but FDM friction, GC3 stalls at $\ln\Lambda$6–$\ln\Lambda$7 after $\ln\Lambda$8. The same study predicts that FDM suppresses in-situ/ex-situ mixing of globular clusters in dwarfs and can produce a bimodal radial distribution for $\ln\Lambda$9, yielding a specific observational target for Euclid DR1 and related extragalactic globular-cluster surveys (Szpilfidel et al., 1 Oct 2025).

The Large Magellanic Cloud wake probes a different, faster-perturber regime. Windtunnel-style simulations with $$i\hbar\,\partial_t\psi = -\frac{\hbar^2}{2m_a}\nabla^2\psi + m_a\Phi\,\psi, \qquad \nabla^2\Phi = 4\pi G\,\rho, \qquad \rho = m_a|\psi|^2,$$0 show that the time-averaged FDM drag on the LMC closely matches the classical Chandrasekhar expectation and the CDM runs with wake self-gravity included. However, the FDM wake is more granular and $$i\hbar\,\partial_t\psi = -\frac{\hbar^2}{2m_a}\nabla^2\psi + m_a\Phi\,\psi, \qquad \nabla^2\Phi = 4\pi G\,\rho, \qquad \rho = m_a|\psi|^2,$$1 dynamically colder than the CDM wake, while the stellar halo responds with percent-level additional heating. In the fiducial case, the stellar radial-velocity-dispersion enhancement is $$i\hbar\,\partial_t\psi = -\frac{\hbar^2}{2m_a}\nabla^2\psi + m_a\Phi\,\psi, \qquad \nabla^2\Phi = 4\pi G\,\rho, \qquad \rho = m_a|\psi|^2,$$2 in FDM versus $$i\hbar\,\partial_t\psi = -\frac{\hbar^2}{2m_a}\nabla^2\psi + m_a\Phi\,\psi, \qquad \nabla^2\Phi = 4\pi G\,\rho, \qquad \rho = m_a|\psi|^2,$$3 in CDM with self-gravity and $$i\hbar\,\partial_t\psi = -\frac{\hbar^2}{2m_a}\nabla^2\psi + m_a\Phi\,\psi, \qquad \nabla^2\Phi = 4\pi G\,\rho, \qquad \rho = m_a|\psi|^2,$$4 without it. This establishes that FDM does not generically imply weaker net drag; in this regime the main discriminants are wake morphology and kinematics rather than the orbit-averaged force magnitude (Foote et al., 2023).

These application-driven studies also define the main open issues. Analytic theory is validated only in restricted parameter ranges, with transient wave structures becoming important when

$$i\hbar\,\partial_t\psi = -\frac{\hbar^2}{2m_a}\nabla^2\psi + m_a\Phi\,\psi, \qquad \nabla^2\Phi = 4\pi G\,\rho, \qquad \rho = m_a|\psi|^2,$$5

and the Fornax timing problem is no longer solved for $$i\hbar\,\partial_t\psi = -\frac{\hbar^2}{2m_a}\nabla^2\psi + m_a\Phi\,\psi, \qquad \nabla^2\Phi = 4\pi G\,\rho, \qquad \rho = m_a|\psi|^2,$$6 in the analytic treatment (Lancaster et al., 2019). Semi-analytic subhalo models still use CDM-calibrated orbit distributions and Chandrasekhar parameters (Du et al., 2016), while globular-cluster implementations absorb heating into the force coefficient rather than introducing explicit diffusion coefficients (Szpilfidel et al., 1 Oct 2025). A plausible implication is that “FDM dynamical friction” is not a single universal correction to Chandrasekhar drag, but a family of regime-dependent phenomena whose correct description depends on whether the dominant physics is wave-suppressed wake formation, granule-driven diffusion, soliton-core response, or non-linear gravitational cooling.