Geodesic Structure, Thermodynamics and Scalar Perturbations of Mod(A)Max black hole Surrounded by Perfect Fluid Dark Matter

In this work, we investigate the optical properties of a spherically symmetric Mod(A)Max black hole surrounded by perfect fluid dark matter, focusing on key features such as the photon sphere radius, shadow, photon trajectories, and the effective radial force experienced by photons. We also study the dynamics of massive particles around the black hole, deriving the effective potential and, from it, the specific energy and angular momentum of particles moving in circular orbits of fixed radii is discussed. The conditions for marginally stable circular orbits are analyzed, highlighting how the geometric parameters that modify the spacetime curvature influence both the optical and dynamical features. Furthermore, we explore the thermodynamic behavior of the black hole by examining its temperature, Gibbs free energy, and heat capacity, as well as its thermodynamic topology. Finally, scalar field perturbations are considered through the massless Klein-Gordon equation, and the quasinormal modes (QNMs) in the eikonal regime are computed, illustrating how the geometric parameters affect the potential and the QNM spectra.

💡 Research Summary

The paper investigates a spherically symmetric black‑hole solution that incorporates three distinct ingredients: (i) a perfect‑fluid dark‑matter (PFDM) halo, (ii) a nonlinear electrodynamics sector described by the Mod(A)Max theory, and (iii) a possible phantom sign in the electromagnetic contribution (η = ±1). Starting from the action
(S=\frac{1}{16\pi}\int d^{4}x\sqrt{-g}\bigl(R-4\eta\mathcal L_{\rm Mod(A)Max}\bigr)+S_{\rm PFDM}),
the authors obtain the metric function
(f(r)=1-\frac{2M}{r}+ \eta e^{-\gamma}\frac{Q^{2}}{r^{2}}+\lambda,\frac{\ln|\lambda r|}{r}).
Here (M) is the black‑hole mass, (Q) the electric charge, (\gamma) the dimensionless ModMax parameter, (\lambda) the PFDM coupling, and (\eta=\pm1) distinguishes the ordinary (η=+1) from the phantom (η=−1) branch. The function is asymptotically flat and its zeros determine the horizon radius (r_{h}). The authors illustrate how the sign and magnitude of (\lambda) inflate or deflate the horizon, how increasing (Q) produces an inner‑outer horizon structure reminiscent of Reissner‑Nordström, and how the phantom sign shifts the horizon outward.

Photon sphere and shadow.
For null geodesics the conserved energy (E) and angular momentum (L) lead to the effective potential (V_{\rm eff}=L^{2}f(r)/r^{2}). The photon‑sphere radius (r_{ps}) follows from (\frac{d}{dr}\bigl(f(r)/r^{2}\bigr)=0). This condition is a nonlinear equation containing all four parameters; the authors solve it numerically and map the dependence of (r_{ps}) on ((Q,\gamma,\lambda,\eta)). They find that electric charge and the ModMax nonlinearity tend to shrink the photon sphere, while a negative PFDM parameter ((\lambda<0)) expands it. The phantom branch ((\eta=-1)) reverses the sign of the electromagnetic term and generally yields a larger photon sphere. The critical impact parameter is (b_{c}=r_{ps}/\sqrt{f(r_{ps})}); the corresponding shadow radius observed at distance (D) is (\theta_{sh}=b_{c}/D). Parameter scans reveal that PFDM can modify the shadow size by up to ten percent, a potentially observable effect for Event Horizon Telescope‑scale imaging.

Timelike geodesics and ISCO.
For massive particles the effective potential becomes (V^{(m)}{\rm eff}=f(r)\bigl(1+L^{2}/r^{2}\bigr)). Circular orbits satisfy (dV^{(m)}{\rm eff}/dr=0); marginal stability adds (d^{2}V^{(m)}{\rm eff}/dr^{2}=0). The authors derive analytic expressions for the specific energy and angular momentum of circular orbits and compute the innermost stable circular orbit (ISCO) radius (r{\rm isco}) as a function of the model parameters. The trends mirror those of the photon sphere: larger charge and stronger ModMax nonlinearity pull the ISCO inward, while negative PFDM pushes it outward. The phantom case again yields the most outward ISCO, implying a reduced accretion efficiency.

Thermodynamics.
Using the horizon radius, the Hawking temperature is (T_{H}=f’(r_{h})/(4\pi)). The entropy follows the area law (S=\pi r_{h}^{2}). From these, the Gibbs free energy (G=M-TS) and the heat capacity (C=\partial M/\partial T) are obtained. The heat capacity changes sign at critical values of (\lambda) and (\gamma), signalling local thermodynamic stability/instability transitions. In the phantom branch the temperature can become negative, indicating a pathological regime. The authors also apply a recent thermodynamic‑topology method (vector‑field zero analysis) to map the phase‑transition structure, showing that PFDM and ModMax parameters shift the location and nature of the topological defects that correspond to phase changes.

Scalar perturbations and quasinormal modes.
The massless Klein‑Gordon equation (\Box\Phi=0) is separated using spherical harmonics, leading to a radial wave equation (\frac{d^{2}\psi}{dr_{*}^{2}}+