Dehnen Method in Stellar and Halo Modeling

• Dehnen Method is a profile-based approach that employs Dehnen-type density laws to model self-gravitating stellar systems and dark-matter environments with varying inner slopes.
• It integrates these density profiles into simulations and analytic calculations, such as N-body dynamics in star clusters and perturbative techniques in relativistic black-hole frameworks.
• The method reveals practical insights on orbit shifts, waveform phase drifts in EMRI studies, and spectral modifications in ringdown analyses, underscoring its broad astrophysical impact.

Dehnen Method denotes, in the literature considered here, not a single algorithmic procedure but the use of Dehnen or Dehnen-type density profiles as structural ansätze for self-gravitating stellar systems and dark-matter environments, followed by the propagation of those profiles into dynamical, relativistic, and perturbative calculations. In star-cluster dynamics, this means prescribing a Dehnen stellar density law, reconstructing the residual gas under constant star-formation efficiency per free-fall time, and then evolving the system after instantaneous gas expulsion with direct $N$-body methods (Shukirgaliyev et al., 2021). In relativistic applications, it means embedding Schwarzschild-like or Kerr-like geometries in Dehnen-type halos, or sourcing the metric directly by Dehnen-type matter, and then studying horizons, geodesics, shadows, EMRIs, quasinormal modes, scattering, or neutrino flavor oscillations (Shokirov et al., 1 Jul 2026, Meng et al., 27 Apr 2026, Bolokhov, 17 Nov 2025).

1. Terminology and profile structure

A recurrent point in this literature is that “Dehnen” usually refers to a density-profile family, not to a numerical integrator or perturbative trick. In the star-cluster formulation, the stellar density is written as

$$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$

with $0\le \gamma<3$, inner behavior $\rho_D(r)\propto r^{-\gamma}$ for $r\ll a_\mathrm{D}$, and outer behavior $\rho_D(r)\propto r^{-4}$ for $r\gg a_\mathrm{D}$ (Shukirgaliyev et al., 2021). The associated half-mass radius is

$$r_\mathrm{h}=a_\mathrm{D}\left(2^{1/(3-\gamma)}-1\right)^{-1},$$

so $\gamma$ changes the inner concentration while the outer slope remains fixed (Shukirgaliyev et al., 2021).

In halo applications, the generalized Dehnen/Zhao form appears as the $(1,4,\gamma)$ family,

$$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$0

with enclosed mass

$$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$1

This makes the cored case $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$2, the Hernquist case $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$3, and the Jaffe case $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$4 special members of a single finite-mass family with fixed outer falloff $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$5 (Shokirov et al., 1 Jul 2026).

Several black-hole papers, however, do not use the full family in this explicit form. One static neutrino-oscillation study introduces only a specific effective metric function,

$$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$6

and identifies $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$7 and $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$8 merely as the radius and density of the dark-matter halo, with the Schwarzschild limit recovered at $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$9 (Alloqulov et al., 23 Oct 2025). Likewise, the rotating-EMRI analysis works directly with a Kerr-like metric containing a radius-dependent effective mass function $0\le \gamma<3$0, rather than reprinting the standard Dehnen density in Newtonian form (Meng et al., 27 Apr 2026). A further specialization, widely used in cored-halo models, fixes $0\le \gamma<3$1, giving

$0\le \gamma<3$2

which is finite at the center and falls as $0\le \gamma<3$3 at large radius (Al-Badawi et al., 26 Jan 2025).

2. Stellar-dynamical formulation in embedded clusters

In the cluster-survival problem, the Dehnen method is a complete workflow rather than a profile choice in isolation. The procedure begins by choosing either a Plummer stellar profile or a Dehnen stellar profile with fixed final stellar mass $0\le \gamma<3$4 and stellar half-mass radius $0\le \gamma<3$5. It then reconstructs the residual gas distribution under the assumption of constant efficiency per free-fall time,

$0\le \gamma<3$6

so that denser regions convert gas into stars more quickly and the residual gas profile becomes shallower than the stellar one. This yields a centrally peaked local SFE profile (Shukirgaliyev et al., 2021).

The local and cumulative efficiencies are defined as

$0\le \gamma<3$7

with total efficiency

$0\le \gamma<3$8

For a given stellar density $0\le \gamma<3$9 and product $\rho_D(r)\propto r^{-\gamma}$0, the residual gas density is recovered analytically via the inversion formulas reproduced from Bek et al.; the authors emphasize that the gas profile depends only on the product $\rho_D(r)\propto r^{-\gamma}$1, not on $\rho_D(r)\propto r^{-\gamma}$2 and $\rho_D(r)\propto r^{-\gamma}$3 separately (Shukirgaliyev et al., 2021).

The stellar system is then placed in virial equilibrium within the combined stellar-plus-gas potential before gas expulsion. Realizations are generated with new acceleration plug-ins for the NEMO/falcON toolchain and with Agama, and the subsequent direct $\rho_D(r)\propto r^{-\gamma}$4-body evolution is carried out with $\rho_D(r)\propto r^{-\gamma}$5-GRAPE/GPU, a fourth-order Hermite integrator using GPU/CUDA GRAPE emulation. Stellar evolution is included through an updated sse implementation, natal kicks of supernova remnants are included, binaries are omitted, and a small softening $\rho_D(r)\propto r^{-\gamma}$6 in $\rho_D(r)\propto r^{-\gamma}$7-body units is used (Shukirgaliyev et al., 2021).

Gas expulsion is instantaneous: the gas potential is removed abruptly at $\rho_D(r)\propto r^{-\gamma}$8, and the post-expulsion violent relaxation is followed for 150 Myr, with

$\rho_D(r)\propto r^{-\gamma}$9

adopted as the end of violent relaxation. The retained mass measure is the Jacobi-mass fraction

$r\ll a_\mathrm{D}$0

and the paper concludes that violent relaxation lasts no longer than $r\ll a_\mathrm{D}$1 Myr independently of density-profile slopes (Shukirgaliyev et al., 2021).

The central result is that Dehnen clusters survive gas expulsion much better than Plummer clusters when SFE is defined in physically comparable apertures such as the Jacobi radius or the half-mass radius. For re-simulated Plummer models, the minimum survival threshold is about $r\ll a_\mathrm{D}$2 when measured within $r\ll a_\mathrm{D}$3, or about $r\ll a_\mathrm{D}$4 within the Jacobi radius; the corresponding thresholds are about $r\ll a_\mathrm{D}$5 and $r\ll a_\mathrm{D}$6. For the Dehnen $r\ll a_\mathrm{D}$7 model, the minimum SFE needed to survive is about $r\ll a_\mathrm{D}$8 if measured within $r\ll a_\mathrm{D}$9, or about $\rho_D(r)\propto r^{-4}$0 within $\rho_D(r)\propto r^{-4}$1, and for steeper cusps the minimum needed to survive can be as low as $\rho_D(r)\propto r^{-4}$2 for $\rho_D(r)\propto r^{-4}$3 and $\rho_D(r)\propto r^{-4}$4 (Shukirgaliyev et al., 2021).

This formulation also clarifies a methodological caveat. Because the reconstructed gas profile can be shallow enough that its cumulative mass diverges if extended to infinity, “global SFE” is aperture-dependent. The paper therefore contrasts $\rho_D(r)\propto r^{-4}$5, $\rho_D(r)\propto r^{-4}$6, the local stellar fraction at $\rho_D(r)\propto r^{-4}$7, and the effective SFE $\rho_D(r)\propto r^{-4}$8, and stresses that scale-radius-based apertures are not equivalent across Plummer and Dehnen families (Shukirgaliyev et al., 2021).

3. Static halo-dressed black holes: metrics, geodesics, thermodynamics, and shadows

A large relativistic branch of the Dehnen method begins from the static spherical line element

$\rho_D(r)\propto r^{-4}$9

with the halo contribution absorbed into $r\gg a_\mathrm{D}$0. In the generalized $r\gg a_\mathrm{D}$1 halo model, the lapse is

$r\gg a_\mathrm{D}$2

which corresponds to the enclosed dark-matter mass $r\gg a_\mathrm{D}$3 of the generalized Dehnen family. Timelike geodesics then follow from

$r\gg a_\mathrm{D}$4

and the numerical analysis shows that increasing $r\gg a_\mathrm{D}$5, $r\gg a_\mathrm{D}$6, or $r\gg a_\mathrm{D}$7 pushes the marginally bound orbit and ISCO to larger radii and angular momenta, raises the ISCO energy for fixed $r\gg a_\mathrm{D}$8 when $r\gg a_\mathrm{D}$9 is larger, enlarges periodic orbits, lengthens orbital periods, lowers waveform amplitudes, and shifts spectral peaks to lower frequencies. In the examples considered, the gravitational-wave spectra remain mainly in the millihertz band, and several characteristic-strain peaks lie above the sensitivity curves of LISA, Taiji, and TianQin (Shokirov et al., 1 Jul 2026).

A cored Dehnen-$$r_\mathrm{h}=a_\mathrm{D}\left(2^{1/(3-\gamma)}-1\right)^{-1},$$0 specialization combined with a Kiselev-type quintessence term uses

$$r_\mathrm{h}=a_\mathrm{D}\left(2^{1/(3-\gamma)}-1\right)^{-1},$$1

and frequently adopts $$r_\mathrm{h}=a_\mathrm{D}\left(2^{1/(3-\gamma)}-1\right)^{-1},$$2, so that the quintessence contribution becomes $$r_\mathrm{h}=a_\mathrm{D}\left(2^{1/(3-\gamma)}-1\right)^{-1},$$3. This model has two horizons, an event horizon and a cosmological horizon. The reported trends are that increasing $$r_\mathrm{h}=a_\mathrm{D}\left(2^{1/(3-\gamma)}-1\right)^{-1},$$4, $$r_\mathrm{h}=a_\mathrm{D}\left(2^{1/(3-\gamma)}-1\right)^{-1},$$5, or the quintessence parameter increases the event horizon and decreases the cosmological horizon, enlarges the photon sphere and shadow, increases the weak deflection angle obtained via the Gauss–Bonnet theorem, and lowers both the oscillation frequency and the decay rate of quasinormal modes relative to vacuum Schwarzschild (Al-Badawi et al., 26 Jan 2025).

Another static realization uses a Dehnen-$$r_\mathrm{h}=a_\mathrm{D}\left(2^{1/(3-\gamma)}-1\right)^{-1},$$6 halo, with

$$r_\mathrm{h}=a_\mathrm{D}\left(2^{1/(3-\gamma)}-1\right)^{-1},$$7

or equivalently

$$r_\mathrm{h}=a_\mathrm{D}\left(2^{1/(3-\gamma)}-1\right)^{-1},$$8

In that study, M87$$r_\mathrm{h}=a_\mathrm{D}\left(2^{1/(3-\gamma)}-1\right)^{-1},$$9 shadow data are used to constrain the halo parameters through the $\gamma$0 interval

$\gamma$1

The allowed region shows a negative correlation between $\gamma$2 and $\gamma$3: larger $\gamma$4 requires smaller $\gamma$5, and conversely. Within the allowed parameter sets, increasing either halo parameter shifts the event horizon, photon sphere, and shadow radius outward, lowers the peak of the scalar, electromagnetic, and axial-gravitational effective potentials, and reduces both $\gamma$6 and $\gamma$7, so the oscillation frequency decreases and damping becomes weaker (Liang et al., 21 May 2025).

A more formal halo-to-metric construction appears in the Dehnen $\gamma$8 exact black-hole solution. The procedure is explicit: choose

$\gamma$9

integrate it to

$(1,4,\gamma)$0

infer a circular velocity $(1,4,\gamma)$1, reconstruct a halo redshift factor from

$(1,4,\gamma)$2

and then superpose a Schwarzschild term to obtain

$(1,4,\gamma)$3

The entropy remains the area law $(1,4,\gamma)$4, but the halo changes the Hawking temperature, heat capacity, and Gibbs free energy, and can generate a positive-heat-capacity branch and second-order phase transitions. A central caveat is that

$(1,4,\gamma)$5

so the spacetime is not asymptotically flat in the standard normalized sense (Senjaya et al., 3 Feb 2026).

4. Rotating Dehnen black holes and EMRI modelling

Rotation is typically introduced by a Newman–Janis-type construction applied to a Dehnen-dressed static seed. In the rotating Dehnen-$(1,4,\gamma)$6 galactic-halo model, the density is

$(1,4,\gamma)$7

the seed lapse is

$(1,4,\gamma)$8

and the modified Newman–Janis algorithm in the Azreg-Aïnou form yields a Kerr-like metric with

$(1,4,\gamma)$9

Horizons are determined by $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$00, the stationary limit surface by $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$01, and the geodesics remain Hamilton–Jacobi separable. The reported phenomenology is Kerr-like in structure but halo-sensitive in scale: increasing $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$02 or $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$03 pushes the horizons and ergosurfaces outward, enlarges the ergoregion, increases the shadow size, and slightly reduces the distortion parameter $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$04, approximately linearly with $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$05 and nonlinearly with $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$06 (Gohain et al., 25 Aug 2025).

The most developed perturbative use of a rotating Dehnen halo appears in EMRI modelling. There the metric is written in Boyer–Lindquist form with

$$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$07

where the effective mass function $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$08 encodes the halo, and the halo parameters are a mass parameter $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$09, a scale radius $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$10, and a Dehnen-profile parameter $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$11, with $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$12 cored and $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$13 cuspy. The spacetime remains Petrov type D, so the Newman–Penrose and Teukolsky machinery can still be used. Variable separation gives an angular equation identical to Kerr and a radial equation modified by the Dehnen halo through $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$14, $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$15, $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$16, or equivalently $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$17 and its derivatives (Meng et al., 27 Apr 2026).

For numerical work, the long-range Teukolsky-type radial equation is transformed to Sasaki–Nakamura form. The halo then enters the geodesic potentials, the harmonic frequencies

$$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$18

the amplitudes $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$19 and $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$20, the horizon flux coefficients, and the reconstructed polarizations

$$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$21

The numerical conclusions are that increasing $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$22 reduces waveform amplitude, increases phase drift, and raises the mismatch relative to Kerr; the mismatch also increases with spin $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$23. For a fixed $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$24, the waveforms become distinguishable once

$$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$25

because the mismatch exceeds the LISA-like threshold

$$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$26

for $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$27 and $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$28. The mismatch is reported to be relatively insensitive to whether the profile is cored or cuspy, so in this parameter study the halo mass parameter and black-hole spin dominate over the $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$29 versus $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$30 distinction (Meng et al., 27 Apr 2026).

5. Dehnen-supported regular black holes and perturbative spectra

A distinct branch of the Dehnen method treats the profile not as an external halo correction to a vacuum black hole, but as the matter source of a regular black-hole geometry. The canonical example is the asymptotically flat lapse

$$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$31

where $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$32 is the ADM mass and $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$33 is the characteristic scale of the surrounding matter distribution. This metric is associated with the specific Dehnen-type density choice

$$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$34

equivalently the $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$35 case in the generalized halo notation. The model is Schwarzschild in the limit $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$36, asymptotically Schwarzschild for $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$37, and de Sitter-like near the origin because

$$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$38

so the central singularity is replaced by a regular core (Bolokhov, 17 Nov 2025).

For scalar, electromagnetic, and Dirac perturbations, the master equation is

$$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$39

with single-barrier effective potentials. Quasinormal frequencies are computed with sixth- and ninth-order WKB methods with Padé corrections and checked by time-domain integration. The agreement is reported as excellent. Across the test-field sectors, increasing $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$40 produces a moderate monotonic increase in $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$41 while leaving the damping rate almost unchanged or only slightly modified. The paper concludes that appreciable deviations from Schwarzschild require unrealistically dense halos, so ordinary galactic dark-matter environments are not expected to alter astrophysical ringdown appreciably (Bolokhov, 17 Nov 2025).

In the gravitational axial sector, the matter-supported background induces a non-vacuum structural novelty. Because the source is an anisotropic fluid, there are two consistent axial perturbation prescriptions, denoted “up” and “down,” with potentials

$$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$42

and

$$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$43

Because $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$44, the two channels are not isospectral. An analytic expansion beyond the eikonal regime gives compact formulas for peak positions and quasinormal frequencies through order $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$45 and inverse powers of $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$46. The reported qualitative trend is that $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$47 grows monotonically with $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$48 for both channels, while the imaginary part changes more mildly (Malik, 19 Mar 2026).

Wave propagation studies of the same regular geometry extend the analysis to grey-body factors, absorption cross-sections, and late-time tails. The gravitational QNM spectrum is again computed in both “up” and “down” sectors with WKB and checked by time-domain integration, confirming the spectral splitting. The late-time decay remains the Schwarzschild Price law,

$$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$49

so for $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$50 one has $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$51. A noted textual tension in this literature is that one wave-propagation study gives partially inconsistent prose descriptions of whether increasing $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$52 raises or lowers the barrier, even though its reported numerical trends indicate reduced absorption for nonzero $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$53 and sector-dependent transmission. This does not affect the paper’s main structural claim that the halo breaks vacuum isospectrality while leaving the asymptotic tail exponent unchanged (Lütfüoğlu et al., 27 Nov 2025).

6. Neutrino phases, observability, and conceptual limits

The Dehnen method has also been used outside gravitational-wave and geodesic problems, most explicitly in neutrino propagation through a Schwarzschild black hole surrounded by a Dehnen-type halo. There the metric is again static and spherical,

$$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$54

with

$$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$55

and on the equatorial plane one has $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$56, $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$57, $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$58, so that

$$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$59

This identity is crucial: for radial propagation, the leading oscillation phase is the same as in Schwarzschild, so the halo is invisible at leading order in that sector. For non-radial propagation, however, the phase, lensing, and damping are modified through the characteristic combinations

$$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$60

and the weak-lensing scale

$$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$61

In the Gaussian wave-packet treatment, the damping factor increases when $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$62 or $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$63 is increased, and the oscillation-probability curves shift toward larger azimuthal angle $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$64 (Alloqulov et al., 23 Oct 2025).

This neutrino application sharpens a broader conceptual point. In several black-hole papers, “Dehnen” does not mean the full general $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$65-family is being carried through explicitly; rather, a specific effective metric is adopted and parameterized only by a density scale and a radius scale. Some studies therefore work with Dehnen-inspired halo metrics without defining the usual inner-slope parameter, characteristic density in standard notation, or the standard enclosed mass $$\rho_D(r)=\frac{(3-\gamma)M_\star}{4\pi}\frac{a_\mathrm{D}}{r^\gamma(r+a_\mathrm{D})^{4-\gamma}},$$66 of the full family (Alloqulov et al., 23 Oct 2025). A plausible implication is that “Dehnen Method” is best understood as a profile-based modelling strategy whose exact content must be read from the specific parametrization actually used.

Across the literature, observability claims are cautious. The neutrino study concludes that Dehnen-halo signatures are measurable only in principle and that such experiments are currently unattainable (Alloqulov et al., 23 Oct 2025). The EMRI study finds a concrete parameter regime in which a LISA-like mismatch criterion is exceeded, so Dehnen-halo effects can become waveform-distinguishable for sufficiently large halo mass parameter and spin (Meng et al., 27 Apr 2026). By contrast, the regular-black-hole ringdown study concludes that noticeable spectral shifts occur only for unrealistically dense halos, so ordinary galactic environments leave quasinormal ringing essentially Schwarzschild-like (Bolokhov, 17 Nov 2025).

The central misconception therefore admits a precise correction. In the cited work, the Dehnen method is not a single numerical technique. It is a family of profile-driven constructions in which a Dehnen or Dehnen-type density law is taken as the primary structural input, and the resulting mass distribution is then propagated into star-cluster survivability, static or rotating black-hole geometries, geodesic observables, quasinormal spectra, EMRI waveforms, scattering data, or neutrino flavor phases. Its scientific content lies in how faithfully that profile choice is translated into the subsequent dynamical or relativistic framework.