Dark Matter Density-Dependent Tunneling

• Dark-Matter-Density-Dependent Tunneling is a phenomenon where ultralight dark matter escapes a subhalo by tunneling through a density-shaped gravitational barrier.
• The research employs both numerical shooting and semiclassical WKB methods to relate core density, host halo properties, and the resulting tunneling rates.
• The analysis establishes survivability bounds for ultralight particles, showing that high core densities suppress tunneling while low densities lead to rapid evaporation.

Searching arXiv for the cited work and closely related ultralight-DM tunneling papers. Dark-matter-density-dependent tunneling denotes the dependence of the escape rate of ultralight dark matter from a self-gravitating subhalo or satellite on the dark matter density profile that sources the gravitational barrier through which the field tunnels. In the ultralight scalar or axion regime, the dark matter is described by a high-occupancy wave field rather than by classical point particles, and tidal stripping by a host halo acquires a genuinely wave-mechanical component analogous to quantum tunneling. In this setting, the central density of the subhalo core, the mean density of the host halo at the orbital radius, and the asphericity of the tidal field jointly determine whether the core is effectively stable or rapidly evaporates (Hertzberg et al., 2022).

1. Definition and physical setting

Dark-matter-density-dependent tunneling arises in ultralight dark matter halos whose de Broglie wavelengths are large enough that particles can leak through a tidal barrier instead of being removed only at a classical tidal radius. Hertzberg and Loeb analyze this effect for dwarf satellites embedded in the tidal field of a host halo, emphasizing that the relevant escape process is tunneling of the scalar field out of the satellite halo (Hertzberg et al., 2022). A later multifield treatment frames the same phenomenon as a standard one-dimensional WKB tunneling problem once the subhalo potential is specified, and extends it to halos composed of more than one ultralight species (Lehmann et al., 12 May 2026).

The density dependence is central. In the single-field treatment, the tunneling rate collapses onto a fit of the form

$$\Gamma \simeq \Gamma_0 \exp[-B(\rho_c/\bar{\rho})],$$

with $\bar{\rho}=3\omega^2/(4\pi G)$ the mean host-halo density within the orbital radius, $\Gamma_0 \simeq 0.77\sqrt{\beta}\,\omega$, $B \simeq 0.23/\beta$, and $\beta \approx 2+\gamma$ (Hertzberg et al., 2022). In the semiclassical formulation, the same qualitative dependence appears as

$$\Gamma_{\rm phys}\sim \rho_0^{1/2}\exp[-\alpha(\rho_h/\rho_0)],$$

so that small $\rho_0$ corresponds to rapid tunneling, whereas large $\rho_0$ yields exponentially suppressed mass loss (Lehmann et al., 12 May 2026). This suggests that the topic is not merely “tidal stripping” in the conventional sense, but a density-controlled barrier-penetration problem in a self-consistent gravitational potential.

2. Field-theoretic formulation and reduction to Schrödinger–Poisson

The single-field formulation begins from a real scalar $\phi$ of mass $m$ coupled to gravity in the nonrelativistic, Newtonian gauge. Writing

$\bar{\rho}=3\omega^2/(4\pi G)$0

and taking the high-occupancy limit, one obtains the effective action

$\bar{\rho}=3\omega^2/(4\pi G)$1

Variation yields the coupled Schrödinger–Poisson equations

$\bar{\rho}=3\omega^2/(4\pi G)$2

and, in a stationary ansatz $\bar{\rho}=3\omega^2/(4\pi G)$3,

$\bar{\rho}=3\omega^2/(4\pi G)$4

To reduce the problem, one expands in spherical harmonics and factors out the $\bar{\rho}=3\omega^2/(4\pi G)$5 piece through $\bar{\rho}=3\omega^2/(4\pi G)$6 (Hertzberg et al., 2022).

The multifield generalization retains the same structure but promotes the potential to one sourced by the sum of species densities. If species $\bar{\rho}=3\omega^2/(4\pi G)$7 has mass $\bar{\rho}=3\omega^2/(4\pi G)$8 and radial wave function $\bar{\rho}=3\omega^2/(4\pi G)$9, then

$\Gamma_0 \simeq 0.77\sqrt{\beta}\,\omega$0

while each field satisfies

$\Gamma_0 \simeq 0.77\sqrt{\beta}\,\omega$1

with $\Gamma_0 \simeq 0.77\sqrt{\beta}\,\omega$2 (Lehmann et al., 12 May 2026). A plausible implication is that density dependence is intrinsically collective in multifield halos, because the barrier seen by one field depends on the density profile of the others through the common Poisson source.

3. Tidal potential, asphericity, and the tunneling channel

For a dwarf on a circular orbit of radius $\Gamma_0 \simeq 0.77\sqrt{\beta}\,\omega$3 around a spherically symmetric host halo with potential $\Gamma_0 \simeq 0.77\sqrt{\beta}\,\omega$4, the host potential can be expanded about $\Gamma_0 \simeq 0.77\sqrt{\beta}\,\omega$5 to second order in a co-rotating frame with angular frequency $\Gamma_0 \simeq 0.77\sqrt{\beta}\,\omega$6. After removing constant and linear terms and re-phasing the wave function, the residual tidal potential is

$\Gamma_0 \simeq 0.77\sqrt{\beta}\,\omega$7

where

$\Gamma_0 \simeq 0.77\sqrt{\beta}\,\omega$8

The parameter $\Gamma_0 \simeq 0.77\sqrt{\beta}\,\omega$9 equals unity for a point-mass halo and is $B \simeq 0.23/\beta$0 for an extended NFW profile (Hertzberg et al., 2022).

The sign structure of $B \simeq 0.23/\beta$1 is decisive. The tidal term opens along the halo–dwarf axis and closes in the orthogonal plane, producing a true tunneling channel rather than isotropic escape (Hertzberg et al., 2022). In the multifield semiclassical description, the effective one-dimensional Hamiltonian is written as

$B \simeq 0.23/\beta$2

so the external tidal field enters as a negative quadratic term competing against the subhalo’s self-gravitational well (Lehmann et al., 12 May 2026).

This asphericity is important for interpreting “density dependence.” The density profile determines the depth and shape of $B \simeq 0.23/\beta$3, while the host environment sets the deformation of the barrier. Dark-matter-density-dependent tunneling therefore depends on density in two distinct senses: the internal central density of the subhalo and the mean density of the host halo at the orbital radius. The 2022 analysis makes this explicit through $B \simeq 0.23/\beta$4, while the 2026 treatment does so through $B \simeq 0.23/\beta$5.

4. Radial reduction and WKB description of the escape rate

To capture the quadrupolar tidal term, the angular decomposition may be truncated at $B \simeq 0.23/\beta$6, yielding two coupled ordinary differential equations for the radial profiles $B \simeq 0.23/\beta$7 and $B \simeq 0.23/\beta$8, together with Poisson’s equation sourced by $B \simeq 0.23/\beta$9 (Hertzberg et al., 2022). In the weak-field limit, the $\beta \approx 2+\gamma$0 piece is negligible inside the core radius, whereas for radii much larger than the core radius the self-gravity term becomes subdominant and one obtains the asymptotic relation

$\beta \approx 2+\gamma$1

with $\beta \approx 2+\gamma$2 for $\beta \approx 2+\gamma$3 and $\beta \approx 2+\gamma$4 a known number. This motivates a single effective radial equation for $\beta \approx 2+\gamma$5,

$\beta \approx 2+\gamma$6

with regularity at the origin and outgoing oscillatory behavior at large radius (Hertzberg et al., 2022).

The semiclassical treatment expresses the same problem directly in WKB form. A quasistationary state has energy $\beta \approx 2+\gamma$7, where $\beta \approx 2+\gamma$8. If $\beta \approx 2+\gamma$9 and $$\Gamma_{\rm phys}\sim \rho_0^{1/2}\exp[-\alpha(\rho_h/\rho_0)],$$0 are the turning points satisfying $$\Gamma_{\rm phys}\sim \rho_0^{1/2}\exp[-\alpha(\rho_h/\rho_0)],$$1, then the leading-order transmission amplitude gives

$$\Gamma_{\rm phys}\sim \rho_0^{1/2}\exp[-\alpha(\rho_h/\rho_0)],$$2

The exponent is

$$\Gamma_{\rm phys}\sim \rho_0^{1/2}\exp[-\alpha(\rho_h/\rho_0)],$$3

so that

$$\Gamma_{\rm phys}\sim \rho_0^{1/2}\exp[-\alpha(\rho_h/\rho_0)],$$4

and the prefactor is

$$\Gamma_{\rm phys}\sim \rho_0^{1/2}\exp[-\alpha(\rho_h/\rho_0)],$$5

Density dependence enters entirely through $$\Gamma_{\rm phys}\sim \rho_0^{1/2}\exp[-\alpha(\rho_h/\rho_0)],$$6 with $$\Gamma_{\rm phys}\sim \rho_0^{1/2}\exp[-\alpha(\rho_h/\rho_0)],$$7, and therefore through both the turning points and the barrier integral (Lehmann et al., 12 May 2026).

A plausible synthesis of these formulations is that the numerical shooting method and the WKB method encode the same barrier physics at different levels of approximation: the former through complex eigenvalues $$\Gamma_{\rm phys}\sim \rho_0^{1/2}\exp[-\alpha(\rho_h/\rho_0)],$$8, the latter through the action under the forbidden region.

5. Density scalings and parametric dependence

In the single-field analysis, if $$\Gamma_{\rm phys}\sim \rho_0^{1/2}\exp[-\alpha(\rho_h/\rho_0)],$$9, then $\rho_0$0 and the instantaneous mass-loss rate is $\rho_0$1. After accounting for the slow change of soliton size, the effective escape rate is $\rho_0$2 (Hertzberg et al., 2022). Numerical solutions for a variety of $\rho_0$3 and $\rho_0$4 collapse onto

$\rho_0$5

with

$\rho_0$6

The core’s tunneling is therefore exponentially suppressed for large central density $\rho_0$7, but becomes violent if $\rho_0$8 (Hertzberg et al., 2022).

The multifield work restates the scaling in dimensionless variables. When lengths are nondimensionalized by $\rho_0$9 and densities by $\rho_0$0, the decay rate in physical units is

$\rho_0$1

with $\rho_0$2. In the regime of interest,

$\rho_0$3

hence

$\rho_0$4

The analysis states that $\rho_0$5 cancels out of the dimensionless rate; $\rho_0$6 enters implicitly by setting $\rho_0$7 and the soliton core radius (Lehmann et al., 12 May 2026).

These relations clarify a common source of confusion. The tunneling rate is not determined by particle mass alone. Rather, particle mass influences the soliton structure, while the actual escape rate is controlled by the density-defined barrier. This suggests that “mass bounds” in ultralight dark matter are, in practice, survivability bounds derived after mapping from density and core size to particle mass.

6. Soliton structure, survivability bounds, and the Fornax window

The connection between density-dependent tunneling and particle-mass constraints is made through the soliton scalings

$\rho_0$8

with $\rho_0$9 known numerically (Hertzberg et al., 2022). Substituting these relations into the escape-rate condition $\phi$0 converts a density-dependent evaporation criterion into a constraint on the ultralight particle mass for given soliton mass $\phi$1 or radius $\phi$2.

For Fornax-like parameters, $\phi$3, $\phi$4, and $\phi$5, the resulting bounds are

$\phi$6

and

$\phi$7

This yields a narrow window of order $\phi$8 if Fornax’s core is to survive a Hubble time (Hertzberg et al., 2022).

The same work states that if another very low density halo is seen, then it rules out the ultralight scalar as core proposal completely, and further notes that non-condensed particles likely impose an even sharper lower bound (Hertzberg et al., 2022). The 2026 multifield analysis qualifies the single-field picture by showing that, in two-field halos, stability bounds can be somewhat relaxed for particular parameter combinations, but for much of the parameter space the constraints become more stringent (Lehmann et al., 12 May 2026). A plausible implication is that survivability bounds inferred from density-dependent tunneling are model-sensitive: they are sharp in the single-field case, but can shift in either direction once multiple ultralight species share the gravitational well.

7. Multifield halos, extensions, and interpretive issues

The multifield generalization introduces separate tunneling rates for each species,

$\phi$9

where $m$0 and $m$1 solve $m$2 (Lehmann et al., 12 May 2026). The fields are coupled because the total potential is sourced by the sum of all densities. In the notation of the 2026 analysis, each species acquires a modified $m$3, where $m$4 and $m$5, so certain mixtures can be more stable or less stable than the single-field case.

The density profile used in that treatment is a two-zone model consisting of a solitonic core and an NFW envelope:

$m$6

or, to good accuracy, $m$7, together with

$m$8

and total density $m$9 (Lehmann et al., 12 May 2026). The corresponding gravitational potential is

$\bar{\rho}=3\omega^2/(4\pi G)$00

up to an additive constant. In practice one often solves Poisson’s equation directly, $\bar{\rho}=3\omega^2/(4\pi G)$01, numerically with the stated boundary conditions (Lehmann et al., 12 May 2026).

Several interpretive points follow directly from the published analyses. First, tidal stripping in ultralight dark matter is not purely classical; tunneling allows escape beyond the classical tidal-radius intuition. Second, the crucial control variable is the barrier shape in the region between the turning points, not only the core center or outer envelope in isolation. Third, the phenomenon is strongly density dependent but not reducible to a single local density: the relevant ratio involves the subhalo core and the host environment. Finally, the existence of multifield halos means that density-dependent tunneling is a property of the full coupled potential, not merely of an isolated species.