Dehnen-Type Dark Matter Halo

• The Dehnen-type dark matter halo is a flexible, spherically symmetric density profile with tunable inner and outer slopes that distinguishes between cored and cuspy structures.
• Its gravitational influence modifies black hole metrics by altering the event horizon, ISCO, and photon sphere, thereby impacting lensing, shadows, and geodesic dynamics.
• Changes in quasinormal modes and gravitational waveforms within Dehnen halos provide valuable probes for assessing dark matter effects in both strong and weak field regimes.

A Dehnen-type dark matter halo is a widely utilized spherically symmetric density profile for galactic halos, especially suited for modeling the dark matter-dominated central regions of galaxies. Its flexibility via tunable inner and outer slopes allows a cored or cuspy central structure, making it an essential framework for investigating the gravitational environment of black holes embedded in galactic halos. Its astrophysical relevance extends to ordinary galaxies, ultra-faint dwarfs, and even strong- and weak-field gravitational tests, where it measurably impacts horizon structure, geodesic dynamics, lensing, black hole shadows, ringdown spectra, and the broader thermodynamic and entropic properties of self-gravitating systems.

1. Dehnen-Type Density Profile: Definition and Parametrizations

The general Dehnen-type density profile is defined as

$$\rho(r) = \rho_0\,\left(\frac{r}{r_0}\right)^{-\gamma}\left[1 + \left(\frac{r}{r_0}\right)^{\alpha}\right]^{\frac{\gamma-\beta}{\alpha}}$$

with parameters:

• $\rho_0$: central (core) density,
• %%%%1%%%%: scale (core) radius,
• $\alpha$: controls transition sharpness,
• $\beta$: asymptotic outer slope,
• $\gamma$: inner slope.

Key specializations are:

• $(\alpha,\beta,\gamma) = (1,4,0)$: cored Dehnen (flat central profile).
• $(1,4,1)$: Hernquist (mild cusp).
• $(1,4,2)$: Jaffe (steep cusp).
• $(1,4,5/2)$: ultra-steep central cusp (applied in dwarf galaxies, see (Xamidov et al., 17 Jul 2025, Ashoorioon et al., 10 Sep 2025)).

This profile allows continuous interpolation between cored and cuspy halos, which is essential for testing competing models of galactic dark matter structure via astrophysical observables.

2. Composite Black Hole–Dehnen Halo Metrics

The static spacetime of a Schwarzschild(-like) black hole embedded in a Dehnen halo is generalized as: $$ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2(d\theta^2 + \sin^2\theta\,d\phi^2)$$ with $f(r)$ augmented by dark matter terms. Typical forms include:

• For Dehnen-(1,4,0):

$$f(r) = 1 - \frac{2M}{r} - \frac{4\pi r_s^3 (r_s + 2r)\,\rho_s}{3(r_s + r)^2}$$

(Gohain et al., 3 Jul 2024, Jha, 26 Jul 2024).

• For Dehnen-(1,4,5/2):

$$f(r) = 1 - \frac{2M}{r} - 32\pi \rho_s r_s^3 \sqrt{\frac{r + r_s}{r_s^2 r}}$$

(Al-Badawi et al., 2 Nov 2024, Xamidov et al., 17 Jul 2025, Hosseinifar et al., 5 Mar 2025, Ashoorioon et al., 10 Sep 2025).

• With quintessence,

$$f(r) = 1 - \frac{2M}{r} - 32\pi \rho_s r_s^3 \sqrt{\frac{r + r_s}{r_s^2 r}} - \frac{c}{r^{3\omega_q + 1}}$$

(Hamil et al., 24 May 2025, Al-Badawi et al., 26 Jan 2025).

The metric can be generalized to include rotation via the Newman–Janis algorithm, yielding a Kerr-like form where the function $\Delta$ is modified by the Dehnen terms, affecting horizon and ergoregion structures (Pantig et al., 2022, Gohain et al., 25 Aug 2025, Uktamov et al., 30 Aug 2025).

3. Black Hole and Halo Interactions: Geometric and Astrophysical Implications

Horizon, Ergoregion, and Geodesic Structure

• Event horizon, ergoregion, and photon sphere radii all increase with higher $\rho_s$ or $r_s$. For coreless/cuspy profiles, the deviation is more pronounced (Jha, 26 Jul 2024, Al-Badawi et al., 26 Jan 2025).
• ISCO (innermost stable circular orbit) and MBO (marginally bound orbit) shift to larger radii: for a fixed black hole mass, $r_{\text{ISCO}}$ grows as halo parameters increase (Alloqulov et al., 7 Apr 2025, Xamidov et al., 17 Jul 2025).
• Timelike and null geodesic analysis reveals systematic modification of potential barriers, affecting accretion disk inner edges, escape conditions, and lensing (Al-Badawi et al., 2 Nov 2024, Gohain et al., 3 Jul 2024, Alloqulov et al., 7 Apr 2025).

Black Hole Shadow and Lensing

• Shadow radius and photon ring expand with increasing halo parameters. For a rotating BH, the shadow is not only larger but also exhibits enhanced distortion for nonzero spin, with the Dehnen profile amplifying both effects (Gohain et al., 25 Aug 2025, Uktamov et al., 30 Aug 2025).
• The critical impact parameter $b_{ph}$ for the shadow edge increases monotonically with $\rho_s$, $r_s$ (Jha, 26 Jul 2024, Luo et al., 26 May 2025).
• Weak gravitational lensing is enhanced: the deflection angle acquires leading corrections proportional to $r_s^3 \rho_s$ (for $(1,4,0)$), and cubic in $r_s$ (see explicit formulas in (Jha, 26 Jul 2024, Al-Badawi et al., 26 Jan 2025, Hamil et al., 24 May 2025)). For cored versus cuspy cases in ultra-faint dwarfs, weak deflection provides a more sensitive discriminator than the shadow or horizon shift (Pantig et al., 2022).

4. Dynamics and Wave Phenomena: QNMs, Gravitational Waves, and Scalar Perturbations

• The effective potentials for scalar, electromagnetic, and gravitational perturbations flatten and shift downward as $\rho_s$ and $r_s$ increase, lowering the peak barrier (Liang et al., 21 May 2025, Al-Badawi et al., 28 Dec 2024, Jha, 26 Jul 2024).
• Quasinormal mode (QNM) frequencies, computed via high-order WKB or time-domain numerics, decrease in both their real and imaginary parts with increasing dark matter content: oscillations are slower and more weakly damped, implying longer ringdown times (Jha, 26 Jul 2024, Al-Badawi et al., 26 Jan 2025, Liang et al., 21 May 2025, Gohain et al., 25 Aug 2025).
• For scalar perturbations, explicit potentials:

$$V_s(r) = f(r)\left[\frac{\ell(\ell + 1)}{r^2} + \frac{f'(r)}{r}\right]$$

where $f(r)$ encodes the Dehnen halo and possibly quintessence (Hamil et al., 24 May 2025, Al-Badawi et al., 26 Jan 2025).

Gravitational Waveforms and EMRIs

• In extreme mass ratio inspirals (EMRIs), orbital parameters $p$ (semi-latus rectum) and $e$ (eccentricity) decay faster in the presence of a Dehnen halo than in vacuum, due both to changes in the potential and to dynamical friction and DM accretion (Ashoorioon et al., 10 Sep 2025).
• Modulation of zoom-whirl orbits and associated GW signatures is predicted, with the "zoom" phase lengthening and phase shifts accumulating relative to vacuum signals—a potential target for LISA/ Taiji (Alloqulov et al., 7 Apr 2025, Ashoorioon et al., 10 Sep 2025).
• For rotating spacetimes, QNM frequencies and damping rates show non-monotonic dependence on halo parameters (Gohain et al., 25 Aug 2025).

5. Thermodynamics, Entropy, and Phase Structure

• Hawking temperature and free energy acquire explicit dependence on $\rho_s, r_s$: $T_H$ generally decreases while the horizon expands (Gohain et al., 3 Jul 2024, Hosseinifar et al., 5 Mar 2025, Ahmed et al., 13 Sep 2025).
• Phase transitions (specific heat divergences, Hawking–Page transitions) are shifted to larger radii or lower temperatures as DM content rises (Gohain et al., 3 Jul 2024, Ahmed et al., 13 Sep 2025).
• The area law for entropy, $S_{BH} = \pi r_h^2$, remains valid; however, the physical area $r_h$ is larger in the presence of a Dehnen halo (Gohain et al., 3 Jul 2024).
• For the dark matter halo itself, the entropic content (in the Verlinde–entropic gravity picture) can be evaluated:

$$S(r) = \frac{2\pi}{5-\gamma} r_0^2 \left(\frac{r}{r_0}\right)^{5-\gamma} {}_2F_1(3-\gamma,5-\gamma;6-\gamma; -r/r_0)$$

for a double power-law Dehnen profile, with closure under the second law (monotonic entropy growth) verified for physically plausible $\gamma$ (Benkrane, 3 Jul 2025).

6. Observational Constraints and Astrophysical Signatures

• Observational data from the Event Horizon Telescope (EHT) for M87*, Sgr A* and other sources have been used to place upper bounds on $(\rho_s, r_s)$ via fits to the angular shadow diameter (Luo et al., 26 May 2025, Jha, 26 Jul 2024, Al-Badawi et al., 28 Dec 2024).
• Astrophysical system modeling (mercury perihelion shift, S2 star orbit, QPOs in microquasars) constrains DM parameters across distinct fields—strongly bounded in Solar System, less so near SMBHs (Xamidov et al., 17 Jul 2025).
• The "deviation parameter" $\delta = R_s/3\sqrt{3}M - 1$ quantifies shadow size departure from Schwarzschild and is used to delimit DM parameter space (Jha, 26 Jul 2024, Luo et al., 26 May 2025).
• In the presence of additional fields (quintessence, cloud string), the qualitative effects remain: expansion of event horizon and shadow, reduction in QNM frequencies (Al-Badawi et al., 28 Dec 2024, Al-Badawi et al., 26 Jan 2025, Ahmed et al., 13 Sep 2025).

7. Extensions: Entropy, Wormholes, and Cosmology

• The Dehnen profile is flexible enough to underpin traversable wormhole solutions within $f(\mathcal{R},\mathcal{L}_m,\mathcal{T})$ gravity without pathological discontinuities or strong exotic matter. The exoticity and anisotropy parameters can be tuned, providing models with negative exoticity (favorable) and sufficient stability (Errehymy et al., 22 Jul 2025).
• Entropy-based emergent gravity scenarios and modified Friedmann equations have been constructed from these profiles, yielding consistent (non-negative) entropic growth and cosmological corrections that, in principle, are testable (Benkrane, 3 Jul 2025).
• The effects of a Dehnen halo extend to modifications of neutrino oscillation phases, with direct corrections to both flavor transition and decoherence scales in curved backgrounds—opening the possibility of using neutrinos as probes of DM environments (Alloqulov et al., 23 Oct 2025).

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Table: Representative Observable Effects of Dehnen-Type Parameters

Phenomenon	Effect of Increasing $\rho_s$, $r_s$	Reference Sections
Event horizon	Increases (radius expands)	2, 3, 5
Photon sphere	Increases (moves outward)	3, 4
Shadow radius	Increases (larger angular size)	3, 4, 6
ISCO	Increases (for massive particles)	3
QNM frequencies	Decrease (oscillations slower, longer-lived)	4
Greybody factors	Increase (higher transmission)	6
GW waveform shift	Enhanced inspiral/circularization	4
Lensing angle	Enhanced deflection (stronger bending)	5, 6
Entropy profile	Grows monotonically with $r$	7

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Summary

A Dehnen-type dark matter halo, parameterized by its inner slope, outer slope, density, and core radius, imprints measurable signatures on the geometry and observables of black holes that reside in galactic halos. Its analytic tractability allows explicit modeling of horizon structure, ISCO location, null and timelike geodesics, and radiative phenomena in settings ranging from solar-system tests to strong-field astrophysical environments. Black hole shadows, gravitational wave spectra (QNMs and inspirals), entropy trends, and even neutrino oscillations acquire distinct modifications as functions of the Dehnen parameters. Current and forthcoming astronomical observations can thus empirically constrain Dehnen-type halo parameters and, by extension, sharpen discriminants among core/cusp and alternative dark matter scenarios in both local and cosmological contexts.