PFDM Black Holes: Modifications & Observables

• Perfect Fluid Dark Matter Black Holes are solutions to Einstein's equations that include a dark fluid generating logarithmic modifications in the metric.
• They alter black hole thermodynamics by shifting critical points, affecting phase transitions, and modifying observable properties like shadows and lensing.
• Their framework provides practical insights into dark matter's environmental effects on supermassive black holes and related astrophysical phenomena.

Perfect fluid dark matter black holes are solutions to the Einstein field equations in which a black hole is surrounded by a spacetime-filling anisotropic fluid designed to model the gravitational effects of galactic dark matter halos. The perfect fluid dark matter (PFDM) energy-momentum tensor provides a minimally coupled and analytically tractable way to include environmental dark sector effects near black holes. Such models lead to characteristic logarithmic modifications in the metric functions, influencing black hole thermodynamics, critical phenomena, lensing, shadow observables, orbital dynamics, and quantum properties. These frameworks are compatible with current galactic and astrophysical observations, and admit extensions to charged/rotating black holes, nonlinear electrodynamics, phantom and quintessence backgrounds.

1. Metric Structure and PFDM Parameterization

The spacetime metric for PFDM black holes generalizes the Schwarzschild, Kerr, Reissner-Nordström, and related solutions by introducing an explicit logarithmic radial dependence stemming from the perfect fluid dark matter source. A typical spherically symmetric solution is:

$ds^2 = -f(r)dt^2 + f(r)^{-1}dr^2 + r^2 d\Omega^2$

$$f(r) = 1 - \frac{2M}{r} + \frac{Q^2}{r^2} + \frac{\alpha}{r}\ln\left(\frac{r}{|\alpha|}\right) + \cdots$$

where $M$ is the black hole mass, $Q$ is charge, and $\alpha$ (or $\lambda$) is the PFDM parameter encoding the dark matter halo’s intensity (Xu et al., 2017, Qiao et al., 2022). For rotating (Kerr-like) solutions:

$$\Delta_r = r^2 - 2 M r + a^2 + \alpha r \ln\left(\frac{r}{|\alpha|}\right) \mp \frac{\Lambda}{3} r^2(r^2 + a^2)$$

with spin $a$ and cosmological constant $\Lambda$ (Xu et al., 2017, Haroon et al., 2018).

In modified theories, further matter sources (nonlinear electrodynamics, Yang-Mills) or string clouds are included via additive potentials or higher-power charge terms (Kumar et al., 3 Mar 2025, Ali et al., 3 Sep 2025, Sood et al., 25 Mar 2024).

Limits and Parameter Roles:

• $\alpha \to 0$: recovers vacuum/standard GR black hole.
• $\alpha>0$: more repulsive/diluting (flattening) effect.
• $\alpha<0$: attractive, steepening the effective potential.

2. Black Hole Thermodynamics and Phase Structure

PFDM modifies all aspects of black hole thermodynamics—mass, temperature, entropy, and critical phenomena.

Extended First Law and Smarr Relation: PFDM enters as a thermodynamic variable: $dM = T dS + \psi dQ^2 + V dP + \Xi d\alpha$

$M = 2TS + Q^2\psi -2VP + \Xi \alpha$

where $\psi$ is the conjugate to $Q^2$, $P$ is the pressure ($-\Lambda/8\pi$), and $\Xi$ is conjugate to $\alpha$ (Xu et al., 2016).

Equation of State and Criticality:

A generic form for the equation of state in PFDM backgrounds is: $Q^2 = r_+(a + r_+ + \Lambda r_+^3 - 4\pi r_+^2 T)$ where $a$ is the PFDM parameter, and $r_+$ the horizon radius. Criticality follows from inflection point conditions, yielding $T_c$, $(Q^2)_c$, $(\psi)_c$ as functions of $\alpha$ (or $a$) (Xu et al., 2016, Sood et al., 25 Mar 2024, Ma et al., 6 Jan 2024).

Critical Exponents and Universality:

• All considered PFDM black holes (RN-AdS, Letelier AdS, AdS ABG) exhibit mean-field critical exponents $(\alpha, \beta, \gamma, \delta) = (0, 1/2, 1, 3)$, as in the Van der Waals fluid (Xu et al., 2016, Sood et al., 25 Mar 2024).
• The universal ratio for the equation of state at critical point $\epsilon = (P_c V_c^{1/3}/T_c)$ is shifted (e.g., $\epsilon \simeq 0.28$ in Letelier/PFDM-AdS, not $3/8$), encoding the influence of PFDM and any string/cloud parameters (Sood et al., 25 Mar 2024).

Stability and Phase Transitions:

• Phase transitions correspond to divergences in the specific heat, $C = ({\partial M}/{\partial T})_{X}$.
• PFDM generally shifts the location and nature of critical points; for regular and nonlinear magnetic-charged black holes, it controls the onset of second-order phase transitions and modifies the stability window (Ndongmo et al., 2023, Sood et al., 25 Mar 2024, Kumar et al., 3 Mar 2025).

Model	Thermodynamic Impact of PFDM	Universality Class
RN-AdS, Letelier AdS	Shifts $T_c$, $r_c$	Mean-field (vdW-like)
EH, ABG, Bardeen black hole	Modifies critical points, stability, entropy (with corrections)	Mean-field (modified $P$-$V$ ratio)

3. Optical Observables: Shadows and Photon Spheres

Shadow Radius and Shape:

• In PFDM backgrounds, the photon sphere and hence the shadow radius are determined by: $2f(r_{ph}) - r_{ph} f'(r_{ph}) = 0$ with shadow radius at infinity

$b_{c} = \frac{r_{ph}}{\sqrt{f(r_{ph})}}$

• For Schwarzschild + PFDM, increasing $\alpha$ (for small/intermediate values) leads to shadow radius reduction; above a critical value, the trend reverses (Qiao et al., 2022, Ma et al., 6 Jan 2024, Su et al., 3 Oct 2024). For rotating/charged/spinning black holes, the shadow is additionally distorted, and PFDM modifies both the size and oblateness (Hou et al., 2018, Ali et al., 3 Sep 2025).

Non-monotonicity and Criticality:

• In models such as Letelier AdS PFDM or Euler–Heisenberg PFDM BHs, shadow radius exhibits non-monotonic behavior with $\alpha$ or $\beta$ (PFDM), reflecting underlying phase transition structure (Sood et al., 25 Mar 2024, Su et al., 3 Oct 2024).
• Jumps/discontinuities in $r_{ph}$ or the impact parameter $u_{ps}$ as $T, P$ cross critical values are geometric order parameters with universal exponent $1/2$.

Observational Prospects:

• Deviation of shadow size and distortion from Kerr predictions can be used to constrain the PFDM parameter via mm/sub-mm VLBI (e.g., EHT), provided requisite angular resolution is achieved (Hou et al., 2018, Ali et al., 3 Sep 2025).

Observable	PFDM Effect
Shadow radius	Decreases (small $\alpha$), then increases at large $\alpha$; non-monotonicity
Shadow distortion	Can sharpen or circularize shadow, shrinks for $k<k_0$, grows for $k>k_0$ (Hou et al., 2018)
Photon sphere	Varies with PFDM, coupled to phase transitions (Sood et al., 25 Mar 2024)

4. Gravitational Lensing, Orbit Dynamics, and Quasinormal Modes

Gravitational Deflection:

• The deflection angle acquires log-corrected perturbative and numerical contributions. The asymptotic expansion: $$\alpha \sim \frac{4M}{b} - \frac{\alpha}{b}\left[1-2 \log 2 + 2 \log\left(\frac{b}{|\alpha|}\right)\right] + \cdots$$ demonstrates that increasing PFDM strength typically reduces the deflection angle, and for large enough PFDM, can make it negative (Qiao et al., 2022, Su et al., 3 Oct 2024).

Orbits and ISCO:

• PFDM reduces the ISCO and photon sphere radii: $r_{isco}, r_{ph}$ both decrease as $\alpha$ or $\lambda$ increase (Shaymatov et al., 2020, Heydari-Fard et al., 2022).
• In the presence of magnetic fields, static PFDM black holes can mimic the ISCO of highly spinning Kerr black holes (degeneracy up to $a/M\sim 0.75-0.8$), complicating spin inference (Shaymatov et al., 2020).

Time Delay and Precession:

• PFDM reduces both time delay and precession angles for bound orbits (Su et al., 3 Oct 2024).
• The viable PFDM equation-of-state for fluid models near black holes requires negative pressure to sustain static, regular DM profiles (Xie et al., 22 Jan 2025).

Quasinormal Modes:

• PFDM raises both the real and imaginary part of QNM frequencies: higher oscillation frequencies and faster decay rates (damping) (Hamil et al., 9 Apr 2024). Effects are systematic and increase with $\alpha$.

5. Quantum Corrections, Thermal Fluctuations, and Evolution

Entropy and Corrections:

• At leading order, the Bekenstein-Hawking entropy maintains the area law $S = \pi r_h^2$ even in PFDM-modified black holes (Kumar et al., 7 Aug 2025, Ndongmo et al., 2023).
• Subleading corrections (logarithmic, e.g., $S = S_0 - \beta \log S_0$) become significant for small black holes (small $r_h$ or Planckian regime) (Kumar et al., 7 Aug 2025).

Thermal Stability and Specific Heat:

• PFDM can add multiple sign crossings to the specific heat curve, leading to rich phase structures: e.g., two phase transitions (unstable $\rightarrow$ stable $\rightarrow$ unstable) (Kumar et al., 7 Aug 2025, Ndongmo et al., 2023).
• The stability region shifts with increasing $\alpha$: higher PFDM shrinks the stable region in many models (e.g., Bardeen PFDM) (Zhang et al., 2020).

Evaporation and Lifetime:

• PFDM slows evaporation and allows the existence of long-lived, near-extremal black holes (zero temperature as $r_h \to \lambda$) (Liang et al., 2023).
• The lifetime enhancement is similar to, but not identical with, charged (Reissner–Nordström) black holes; black holes can remain as remnant objects for cosmological timescales.

6. Topological Thermodynamics and Phase Landscape

A topological classification using the off-shell free energy’s vector field demonstrates that for Schwarzschild, Kerr, RN, and Kerr-Newman black holes, the winding and topological numbers are unaffected by PFDM ($W$ remain as in vacuum). However, for Kerr-AdS-PFDM or (static) Hayward black holes in PFDM backgrounds, PFDM can change the topological class (e.g., $W$ shifts from $0$ to $1$) (Rizwan et al., 2023). The inclusion of nonlinear magnetic field, charge, rotation, and AdS asymptotics leads to a variety of new thermodynamically distinct black hole phases and possible phase transitions, with the number and type of topological defects (generation/annihilation points) sensitive to PFDM and field content.

Black Hole	PFDM?	Charge(s)	$W$
Schwarzschild, Kerr	Any	None	-1, 0
Kerr-AdS, Hayward static	Yes	$Q_m$, AdS	1 or 0

7. Astrophysical Context and Observational Signatures

PFDM black holes provide a robust phenomenological framework for modeling the dark sector’s impact on the environments of supermassive black holes in galactic centers:

• Rotation curves: Asymptotically flat, $\alpha$-independent velocities for $\alpha > 0$ explain observed galaxy rotation profiles (Xu et al., 2017).
• Accretion disk spectra: PFDM reduces the ISCO, leading to hotter, more luminous disks with higher cutoff frequencies (Heydari-Fard et al., 2022). For moderate $\alpha$, the spectral changes can be detected in X-ray/UV observations.
• Imaging: The effect of PFDM on direct images (as with the EHT) is small for currently allowed $\alpha$, but with future sub-$\mu$as resolution, the PFDM-induced modifications in shadow size, distortion, and polarization vectors become potentially observable (Hou et al., 2018, Ali et al., 3 Sep 2025).
• Lensing and GW signatures: Modified deflection angles, shadow radii, and QNM frequencies could be used to indirectly constrain PFDM parameters.

References and Further Developments

The analytic and numerical results referenced are primarily based on the collective works (Xu et al., 2016, Haroon et al., 2018, Hou et al., 2018, Shaymatov et al., 2020, Shaymatov et al., 2020, Heydari-Fard et al., 2022, Qiao et al., 2022, Ndongmo et al., 2023, Liang et al., 2023, Rizwan et al., 2023, Ma et al., 6 Jan 2024, Sood et al., 25 Mar 2024, Hamil et al., 9 Apr 2024, Su et al., 3 Oct 2024, Xie et al., 22 Jan 2025, Kumar et al., 3 Mar 2025, Kumar et al., 7 Aug 2025, Ali et al., 3 Sep 2025).

The PFDM formalism continues to be generalized to include coupled dark energy (quintessence, phantom), nonlinear fields (EH, ABG, Yang-Mills), higher dimensions, and time-dependent backgrounds. Despite the analytic simplicity of the PFDM ansatz, the induced phenomenology is rich and tightly coupled to the observable signatures of astrophysical black holes.