Black holes in general relativity coupled with NEDs surrounded by PFDM: thermodynamics, epicyclic oscillations, QPOs, and shadow

Abstract: In this work, we investigate the thermodynamics and motion of neutral test particles around a regular black hole immersed in a perfect fluid dark matter environment. We begin by examining the horizon structure and key thermodynamic properties, with particular emphasis on quantities such as the Hawking temperature and the specific heat capacity. These aspects provide important insight into the stability and physical behavior of the black hole system. We then proceed to analyze the dynamics of neutral test particles using the Hamiltonian formalism, through which we derive the effective potential governing particle motion. Using the effective potential, we further study quasiperiodic oscillations by determining the associated epicyclic frequencies and comparing them with available observational data. Using the observed QPO data of XTE J1550-564, GRO J1655-40, GRS 1915+105, and M82 X-1, we perform a Markov Chain Monte Carlo analysis to constrain the black hole mass, the magnetic charge parameter, the PFDM parameter, and the characteristic orbital radius. Finally, we investigate the black hole shadow and demonstrate how various geometric parameters influence its optical appearance. This analysis highlights the potential observational signatures of such black holes and their surrounding dark matter environment.

• The paper demonstrates that coupling nonlinear electrodynamics with perfect fluid dark matter produces novel thermodynamic behaviors and distinct phase transitions in black holes.
• It employs parameterized modeling and MCMC analysis to show how magnetic charge and PFDM modify ISCO locations, effective potentials, QPO frequencies, and shadow sizes.
• These findings provide actionable insights for testing deviations from general relativity and for guiding multiwavelength observational strategies in astrophysics.

Black Holes with Nonlinear Electrodynamics in Perfect Fluid Dark Matter: Thermodynamics, Dynamics, QPOs, and Shadow

Introduction and Model Framework

The paper "Black holes in general relativity coupled with NEDs surrounded by PFDM: thermodynamics, epicyclic oscillations, QPOs, and shadow" (2604.13140) presents a systematic analysis of static, spherically symmetric black holes arising from general relativity coupled to nonlinear electrodynamics (NED), embedded in a perfect fluid dark matter (PFDM) environment. The metric derives from a specific NED Lagrangian and incorporates a logarithmic correction to model PFDM, parameterized by $\lambda$, alongside a magnetic charge $q$. This class of solutions unifies two crucial lines of research in black hole physics: regular (nonsingular) black hole models via NED, and environmental effects due to dark matter halos.

The work addresses thermodynamic structure, test particle dynamics, epicyclic and quasi-periodic oscillatory signatures, statistical constraints from observed QPOs, and shadow phenomena. This comprehensive analysis reveals parametric dependencies and potential observational discriminants for black holes in astrophysically realistic, non-vacuum contexts.

Horizon Structure and Thermodynamics

The lapse function $f(r)$ depends on mass $M$, charge $q$, and PFDM parameter $\lambda$. Horizon locations are roots of $f(r)=0$. Increasing either $q$ or $\lambda$ systematically reduces the event horizon radius.

Figure 1: Behavior of the metric function for different values of $\lambda$ and $q$0.

Thermodynamic properties are substantially altered compared to Schwarzschild. The Hawking temperature $q$1 exhibits a non-monotonic dependence on horizon radius $q$2 in the presence of $q$3 or $q$4. There exists a peak temperature, after which $q$5 decreases, unlike the strictly decreasing Schwarzschild profile.

Figure 2: Hawking temperature $q$6 vs. event horizon radius $q$7. Modified solutions demonstrate a peak, lower maximal temperatures compared to Schwarzschild, and return to Schwarzschild-like behavior at large $q$8.

Heat capacity $q$9 undergoes sign changes and divergences at critical radii, marking second-order phase transitions and delineating stable ($f(r)$0) and unstable ($f(r)$1) thermodynamic branches. This is in stark contrast to Schwarzschild, which lacks such thermodynamically stable regimes.

Figure 3: Heat capacity $f(r)$2 as a function of $f(r)$3, showing divergences and a sign change indicative of phase transitions absent in Schwarzschild.

These features signal a fundamentally enriched thermodynamic phase structure due to the NED and PFDM contributions, with possible implications for small black hole remnants and nontrivial endpoint configurations.

Test Particle Motion: Effective Potential, ISCO, and Epicyclic Structure

The dynamics of neutral test particles are encoded by the effective potential $f(r)$4, which is sensitive to $f(r)$5 and $f(r)$6. These parameters shift the location and depth of the potential minimum, resulting in an outward displacement of stable circular orbits and a shallower binding potential.

Figure 4: Effective potential $f(r)$7 as a function of $f(r)$8 under varying PFDM and charge parameters.

The effective radial force also displays a pronounced dependence on $f(r)$9 and $M$0, modulating the strength and the spatial profile of gravitational attraction/repulsion for test particles.

Figure 5: Radial force as a function of $M$1 for representative values of $M$2 and $M$3.

Examination of circular orbits identifies the innermost stable circular orbit (ISCO), where both $M$4 and $M$5 reduce the ISCO radius relative to Schwarzschild, potentially shifting the inner edge of accretion disks and affecting high-energy emission.

Figure 6: Three-dimensional plot of ISCO radius as a function of $M$6 and $M$7.

Test particle trajectories, including bound and unbound orbits, confirm significant deviations from Schwarzschild geodesics, especially in the strong-field regime.

Figure 7: Particle trajectory in the $M$8 plane under fixed $M$9 and varying $q$0.

Epicyclic frequencies, crucial for modeling QPOs, show that both orbital and radial frequencies are enhanced by increasing $q$1 and $q$2. This alters the coupling of geodesic motion to observable phenomena in the accretion flow.

QPO Modeling and Bayesian Constraints

The paper applies the relativistic precession (RP) model to interpret twin-peak QPOs in four accreting compact sources, using coordinate frequencies of geodesic motion in the derived spacetime. Both the upper ($q$3) and lower ($q$4) QPO frequencies systematically shift to higher values for increasing $q$5 and $q$6, for a fixed ISCO.

Figure 8: Correlation between QPO frequencies $q$7 and $q$8 in the RP model. Larger $q$9 and $\lambda$0 raise both frequencies compared to Schwarzschild.

A MCMC parameter estimation was carried out to constrain $\lambda$1 using observed QPO data. Posterior distributions for these parameters converge to small, positive $\lambda$2 and narrow intervals for $\lambda$3, with the best-fit QPO-generative radius near $\lambda$4 across sources (stellar and IMBH regimes).

Figure 9: Posterior distributions of $\lambda$5, $\lambda$6, $\lambda$7, and $\lambda$8 from the MCMC fit to QPO data from multiple XRB and ULX sources.

Importantly, the analysis demonstrates not only consistency with independent dynamical mass estimates but also that QPO observations can robustly constrain non-GR parameters—particularly the magnetic charge and dark matter environment, which are otherwise inaccessible to electromagnetic imaging.

Black Hole Shadow and Null Geodesics

The black hole shadow, determined by the unstable photon sphere, exhibits reduced angular size for increasing $\lambda$9 and $f(r)=0$0. The shadow remains nearly circular due to the underlying spherical symmetry but is systematically contracted compared to Schwarzschild.

Figure 10: Shadow profiles in celestial coordinates under varying $f(r)=0$1, $f(r)=0$2, and $f(r)=0$3.

The three-dimensional analysis of photon sphere and shadow radii confirms that the primary reduction in shadow size arises from magnetic charge, with PFDM parameter playing a subdominant environmental role.

Figure 11: 3D dependence of photon sphere and shadow radius on $f(r)=0$4 and $f(r)=0$5.

This makes shadow measurements, such as those performed by the Event Horizon Telescope, a valuable diagnostic for disentangling intrinsic (NED-induced) from environmental (PFDM-induced) effects.

Implications and Future Directions

The paper provides clear numerical evidence for nontrivial thermodynamic phase structure, observable deviations in QPO spectra, and distinct shadow morphologies for black holes in NED with PFDM, relative to pure Schwarzschild backgrounds. The strong correlations among thermodynamic, dynamical, and optical signatures establish the value of this spacetime as a testbed for probing both high-energy physics beyond GR and dark matter phenomenology in the vicinity of black holes.

Practically, the combined approach—leveraging multiwavelength timing (QPOs), gravitational lensing, and horizon-resolving shadow observations—offers a blueprint for constraining non-GR parameters with current and future observational techniques.

Conclusion

The analysis of black holes in nonlinear electrodynamics surrounded by perfect fluid dark matter elucidates the intertwined effects of intrinsic quantum-inspired regularization and environmental dark sector physics. There are robust, observable signatures in thermodynamics, particle dynamics, QPO phenomenology, and shadow properties. This spacetime serves as a unifying framework for connecting strong-field gravity, electromagnetic analogs of quantum corrections, and realistic dark matter environments in astrophysical contexts. Future developments could extend the analysis to rotating solutions and incorporate plasma or further dark sector microphysics to support the next generation of multi-messenger black hole probes.