Perfect Fluid Dark Matter (PFDM)

• PFDM is a phenomenological framework describing dark matter as an isotropic perfect fluid, introducing logarithmic metric corrections in both galactic halos and black hole spacetimes.
• It modifies the Einstein field equations to reproduce flat rotation curves and alter key black hole features such as horizon positions, shadow sizes, and quasinormal mode frequencies.
• PFDM connects local galactic dynamics with global cosmological constraints, influencing thermodynamics, phase transitions, and observational signatures like ISCO shifts and energy emission rates.

Perfect Fluid Dark Matter (PFDM) is a phenomenological framework in which the dark matter component of galactic halos and its influence on astrophysical and black hole spacetimes are modeled using the energy–momentum tensor of an isotropic perfect fluid. PFDM modifies the spacetime geometry through specific metric corrections—most characteristically, logarithmic terms in the metric functions—introducing distinct signatures in gravitational, thermodynamic, and quantum properties of both galaxies and black holes.

1. Formulation and Metric Structure

PFDM is introduced by augmenting the Einstein field equations with a stress–energy tensor of the form $T_{\mu\nu} = (\rho + p)u_\mu u_\nu + p g_{\mu\nu}$, where $\rho$ is the energy density and $p$ the isotropic pressure of the dark matter fluid. In galactic halos, the imposition of flat rotation curves ($v^\phi \sim \text{const}$ at large $r$) yields, via the Einstein equations, a static spherically symmetric metric of the form: $$ds^2 = -B_0 r^l dt^2 + \frac{dr^2}{(c/a) + D/r^a} + r^2(d\theta^2 + \sin^2\theta d\phi^2),$$ with $l = 2(v^\phi)^2$, and the integration constant $D$ corresponding to the background spatial curvature, $D = -k$ (1009.3572).

For compact objects or black holes, PFDM enters as an additive term in the metric function (lapse), typically: $$f(r) = 1 - \frac{2M}{r} + \frac{Q^2}{r^2} + \frac{\alpha}{r}\ln\left(\frac{r}{|\alpha|}\right) + \ldots$$ where $\alpha$ is the PFDM parameter. In rotating spacetimes, the PFDM term enters the horizon equation, for example: $$\Delta = r^2 + a^2 - 2Mr + Q^2 + \alpha r \ln\left(\frac{r}{|\alpha|}\right).$$ This structure is universal across a wide class of black hole solutions, including those incorporating additional matter fields (e.g., nonlinear electrodynamics, Yang–Mills), cosmological constant, or modifications such as the Kalb–Ramond field (Rizwan et al., 2018, Ali et al., 3 Sep 2025, Jha, 27 Jun 2025).

2. Galactic and Cosmological Implications

When the PFDM framework is applied at galactic scales, constancy of the rotation velocity implies nearly pressureless (nonrelativistic) dark matter, with the equation of state given by: $\omega = \frac{p}{\rho} \simeq \frac{l}{4-l}$ where $l \approx 10^{-6}$ for $v^\phi \sim 300$ km/s, yielding $\omega \sim 2.5 \times 10^{-7}$. The halo energy density falls off as $\rho \propto 1/r^2$ as expected for isothermal dark halos. The metric solution embeds the local halo in a static FLRW background, with the spatial curvature parameter $D = -k$, linking the local flatness inferred from galactic dynamics to the global flatness of the universe, consistent with cosmic microwave background constraints (1009.3572).

3. PFDM Effects on Black Hole Spacetimes

3.1 Horizon Structure and Orbits

In black holes, PFDM modifies the positions of event and Cauchy horizons, nontrivially affecting the parameter space for black hole existence. For Kerr-like geometries, an $\alpha$-dependent critical spin $a_c$ demarcates black holes from naked singularities: $a_c = \sqrt{r_e(r_e + \alpha)}$ where $r_e$ is found by extremizing $$a^2(r, \alpha) = 2r - r^2 - \alpha r \ln(r/|\alpha|)$$. For $-2 \leq \alpha < 2/3$, increasing $\alpha$ decreases horizon size, while for $\alpha > 2/3$, the horizon grows. PFDM also impacts the ergoregion size and shifts the innermost stable circular orbit (ISCO), with crucial consequences for the efficiency of processes such as the Penrose mechanism and magnetic reconnection-based energy extraction (Rizwan et al., 2018, Tan et al., 8 Apr 2025, Rodriguez et al., 22 Jul 2024).

3.2 Geodesic Structure and Stability

PFDM-modified metrics yield explicit changes to the effective potential for both massive and massless particles. The critical condition for the photon sphere, $2f(r_p) - r_p f'(r_p) = 0$, is $\alpha$-dependent, as are the stability criteria for circular geodesics through the Lyapunov exponent: $\lambda_p = \sqrt{\frac{V''(r_0)}{2}}.$ The QNM (quasinormal mode) spectrum is similarly modulated, with the real and imaginary parts of the frequencies showing correlated dependencies on PFDM and charge/spin parameters (Das et al., 2023, Tan et al., 8 Apr 2025).

4. Thermodynamics, Phase Structure, and Topology

PFDM fundamentally alters black hole thermodynamics by redefining variables such as mass, temperature, and entropy. The Hawking temperature becomes: $$T_H = \frac{r_h - \lambda}{4\pi r_h^2}, \quad \lambda \sim \text{PFDM parameter},$$ with the requirement $r_h \geq \lambda$ for physical (non-negative) temperature (Liang et al., 2023, Kumar et al., 7 Aug 2025). Accurately capturing the finite-energy character of PFDM metrics requires using the quasi-local Misner–Sharp mass within the dark matter halo radius rather than global (divergent) Komar mass (Liang et al., 2023). Entropy exhibits leading-order area law with subleading logarithmic corrections from thermal fluctuations, crucial in the small-horizon regime (Kumar et al., 7 Aug 2025).

Heat capacity, Gibbs free energy, and phase stability all show dependence on PFDM, often leading to multiple stable and unstable branches and second-order phase transitions. For example, Bardeen, Hayward, and AdS-type black holes with PFDM each display parameter regions where heat capacity diverges, marking phase transition points (Zhang et al., 2020, Ma et al., 2020, Sood et al., 25 Mar 2024).

Universal topological classification of thermodynamic states, based on winding numbers of an augmented free energy current, demonstrates that PFDM shifts but does not fundamentally alter the universal topological class for Schwarzschild, Kerr, or their charged generalizations (with $W \in \{1^-,0^+,0^-,1^+\}$ corresponding to different stability/transition structures) (Rizwan et al., 2023, Rizwan et al., 8 Jan 2025).

5. Photon Orbits, Black Hole Shadows, and Lensing

PFDM leaves observable imprints on null geodesics, shadow radius, and strong lensing observables:

• The shadow’s critical radius $R_s$ and impact parameter $b_m$ depend on the PFDM parameter and can be shifted to larger or smaller values depending on the detailed parameter regime.
• In rotating spacetimes, PFDM increases the deformation and decreases the size of the shadow for fixed spin, with larger PFDM leading to a more prolate or distorted shadow (Ma et al., 2020, Atamurotov et al., 2021, Ali et al., 3 Sep 2025).
• The energy emission rate (e.g., as a Planckian spectrum modulated by $R_s$ and $T_H$) is modified both in peak and frequency, with PFDM generally decreasing the emission height and shifting the peak to lower frequencies (Ma et al., 2020, Ali et al., 3 Sep 2025).
• Lensing coefficients $\bar{a}$ and $\bar{b}$, photon sphere position $x_m$, and observables such as angular separation and magnification in strong lensing, all acquire $\alpha$- and $\beta$-dependent corrections, allowing for the possibility of extracting PFDM constraints from black hole shadow and lensing measurements (e.g., EHT images of M87* and SgrA*) (Atamurotov et al., 2021, Jha, 27 Jun 2025).

6. Quantum Fields, Entanglement, and Coherence in PFDM Backgrounds

Quantum entanglement and coherence properties of matter and radiation fields near PFDM black holes are sensitive to the PFDM parameter $\alpha$. This occurs principally via the PFDM-induced shift to the Hawking temperature $T = (\alpha + r_h)/(2\pi r_h^2)$, impacting the unrecoverable loss of quantum correlations due to Hawking radiation.

• Bosonic fields (scalar modes) show greater degradation in entanglement with increasing PFDM density, while fermionic field coherence is more strongly modulated by PFDM (Wu et al., 22 Jul 2025).
• Susceptibility of entanglement and coherence to PFDM is non-monotonic in $\alpha$, suggesting optimal parameter regimes for quantum dark matter detection strategies depending on the field type and observable (entanglement vs coherence).
• The quantum resource measures, such as the $l_1$-norm of coherence and logarithmic negativity, can be analytically related to the modified surface gravity and thus to PFDM characteristics.

7. Astrophysical and Observational Signatures

PFDM’s metric modifications directly alter the ISCO, shadow size, photon sphere, and emission/absorption rates. Observed properties of accretion disk spectra, QPOs, shadow diameter, lensing images, and gravitational wave ringdowns can all, in principle, be used to constrain the PFDM parameter.

• Black hole energy extraction (Penrose, Blandford–Znajek, magnetic reconnection) for PFDM-environment black holes is enhanced or diminished depending on the ergoregion’s PFDM dependence, with unique parameter regimes allowing high efficiency even away from the extremal spin limit (Rodriguez et al., 22 Jul 2024).
• Shadow and lensing measurements (e.g., deviation $\delta = R_s/(3\sqrt{3} M) - 1$ of the shadow radius from Schwarzschild value) provide direct PFDM constraints when compared with high-resolution VLBI data (Jha, 27 Jun 2025).
• QNM frequencies, governed by the modified effective potential, are also affected in both damping rate and oscillation frequency, offering another channel for indirect PFDM detection in gravitational wave signals (Das et al., 2023, Tan et al., 8 Apr 2025, Pengpan, 15 Aug 2025).

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This synthesis demonstrates that PFDM induces systematic and physically meaningful modifications at galactic, black hole, thermodynamic, and quantum levels. The cumulative effects on metric properties, thermodynamics, lensing, quantum information, and astrophysical processes provide a multi-pronged framework to connect dark matter theory with both astrophysical observations and quantum field theory in curved spacetime.