Shadow, Quasinormal Modes, Sparsity, and Energy Emission Rate of Euler-Heisenberg Black Hole Surrounded by Perfect Fluid Dark Matter

Abstract: In this work, we investigate the optical, dynamical, and radiative properties of an Euler--Heisenberg black hole immersed in a perfect fluid dark matter (PFDM) background. We analyze the photon sphere and shadow, the scalar quasinormal-mode spectrum in the eikonal regime, the grey-body factor through the eikonal QNM correspondence, the sparsity of Hawking radiation, and the corresponding energy emission rate. Our results show that both the black-hole charge and the PFDM parameter significantly affect the photon sphere, shadow size, quasinormal frequencies, Hawking temperature, and emission profile, whereas the Euler--Heisenberg correction is typically subleading in the parameter range explored, although it may become more visible in strong-charge regimes for selected observables. Overall, the dark-matter environment provides the dominant imprint on the phenomenology of the system, indicating that shadow and ringdown-related quantities may serve as useful probes of PFDM effects within the approximations considered.

• The paper establishes that PFDM significantly constrains black hole shadows, quasinormal modes, and Hawking radiation, while Euler–Heisenberg corrections remain subleading for moderate charge.
• It employs null geodesic analysis and eikonal approximations to derive photon sphere properties and energy emission rates under combined gravitational and QED effects.
• The findings suggest that observational signals, such as VLBI shadows and ringdown spectra, can differentiate dark matter influences from intrinsic quantum corrections.

Optical, Dynamical, and Radiative Phenomena of the Euler–Heisenberg Black Hole in Perfect Fluid Dark Matter

Introduction and Theoretical Framework

The paper systematically examines the optical signature, dynamical perturbations, and Hawking radiative behavior of an Euler–Heisenberg (EH) black hole embedded in a perfect fluid dark matter (PFDM) background, extending classical Reissner–Nordström models by incorporating leading-order QED vacuum polarization (through EH nonlinear electrodynamics) and a phenomenological matter halo. The action synthesizes the Einstein-Hilbert gravitational sector, EH-corrected Maxwell dynamics, and a PFDM stress-energy yielding a logarithmic deformation of the spacetime, parameterized by an EH coupling $\alpha$ and PFDM density parameter $\lambda$. The geometry remains asymptotically flat; in limiting regimes ($\alpha \to 0$, $Q \to 0$, $\lambda \to 0$), the metric interpolates to previously studied families (Schwarzschild/charged/PFDM).

Photon Spheres and Black Hole Shadow Properties

The authors obtain the photon sphere radius $r_p$ and shadow radius $R_{\rm sh}$ via null geodesic analysis, with nontrivial modifications due to dark matter and nonlinear electrodynamics. The critical result is that both increasing $|Q|$ and $|\lambda|$ monotonically reduce $r_p$ and $\lambda$0, but the sensitivity to $\lambda$1 is much weaker except in strong-charge regimes. The dark-matter halo introduces a dominant suppression, contracting the observable shadow even at moderate $\lambda$2.

Figure 1: Three-dimensional visualization of the photon sphere radius as a function of $\lambda$3 and $\lambda$4 for two values of $\lambda$5.

Figure 2: Annular photon rings for varying $\lambda$6 and fixed $\lambda$7; an increase in $\lambda$8 expands the size of the rings.

Figure 3: Three-dimensional plot of the black hole shadow radius $\lambda$9, highlighting the parameter dependencies.

Figure 4: Shadow silhouettes in the observer’s celestial plane for various $\alpha \to 0$0, $\alpha \to 0$1, and $\alpha \to 0$2; all lie within the Schwarzschild shadow.

It is emphasized that for most of the parameter space, the EH corrections to the shadow are subleading. However, at large $\alpha \to 0$3 and $\alpha \to 0$4, the effect is visible and distinguishable. The shadow is highly sensitive to the PFDM background, suggesting the surrounding matter profile can in principle be constrained by high-resolution VLBI observations.

Scalar Perturbations and Eikonal Quasinormal Modes

The scalar field perturbation equation in this background produces a modified Regge–Wheeler potential. The effective potential is shown to be mainly deformed by $\alpha \to 0$5, with only minor $\alpha \to 0$6-dependence in most cases.

Figure 5: Effective potential $\alpha \to 0$7 for massless scalar waves as various background parameters are varied.

In the eikonal regime ($\alpha \to 0$8), the quasinormal mode (QNM) spectrum is governed by photon sphere properties via the geodesic correspondence: $\alpha \to 0$9, with $Q \to 0$0 and $Q \to 0$1 being the angular velocity and Lyapunov exponent at $Q \to 0$2. Both quantities grow with $Q \to 0$3 and $Q \to 0$4, but the PFDM parameter is the principal driver. Variations in $Q \to 0$5 can be significant only at high $Q \to 0$6.

Figure 6: Eikonal QNM quantities at fixed $Q \to 0$7, demonstrating the monotonic dependence on $Q \to 0$8 and $Q \to 0$9.

The QNM spectra show that the PFDM environment dominates the real and imaginary components, dictating oscillation frequency and damping rate, respectively. This establishes a hierarchy of phenomenological imprint: $\lambda \to 0$0 in most cases.

Eikonal Grey-Body Factors and Radiative Transport

Using the QNM-Grey Body Factor correspondence, the transmission probability as a function of frequency is derived in the eikonal limit. The transition threshold $\lambda \to 0$1 and the shape of the sigmoid are set mainly by $\lambda \to 0$2 and $\lambda \to 0$3.

Figure 7: Eikonal grey-body factor $\lambda \to 0$4 illustrating the influence of multipole index, $\lambda \to 0$5, and $\lambda \to 0$6 (in high-$\lambda \to 0$7 configurations).

The analysis demonstrates that the PFDM parameter yields the largest shift in the transition threshold and broadening of the sigmoid. Only in strongly charged regimes does $\lambda \to 0$8 manifest as an observable correction to the spectral transmission profile.

Hawking Radiation: Temperature, Sparsity, and Emission Rates

The Hawking temperature follows from the deformed surface gravity, with $\lambda \to 0$9 enhanced by $r_p$0 and decreased by $r_p$1. The sparsity parameter $r_p$2, quantifying deviations from continuous blackbody emission, is invariably much larger than unity across the explored parameter space—confirming that the Hawking cascade is well within the sparse regime.

Figure 8: Hawking temperature as a function of $r_p$3 and $r_p$4 at fixed $r_p$5.

Figure 9: Dimensionless sparsity parameter $r_p$6 characterizing the Hawking emission, as a function of $r_p$7 and $r_p$8.

Energy emission spectra in the geometric-optics limit reflect the combined effect of the shadow (as effective area) and temperature. The flux peak and cutoff frequencies are principally regulated by the PFDM parameter; the impact from EH corrections is negligible except at high charge.

Figure 10: Spectral energy emission rate $r_p$9 illustrating suppression/enhancement by $R_{\rm sh}$0 and $R_{\rm sh}$1 across frequencies.

Implications and Future Directions

This unified treatment demonstrates that the presence of a PFDM background strongly constrains the observable phenomenology of charged black holes, offering observational handles on matter profiles through shadow and ringdown spectra. The EH electrodynamics corrections—though of fundamental theoretical importance for strong-field QED—are subdominant for moderate $R_{\rm sh}$2, and require either extremely high charge or next-generation precision to be disentangled in shadow and emission data.

For gravitational wave astrophysics, both the real and damping frequencies of the ringdown, as well as the temporal structure of Hawking emission, are set primarily by environmental PFDM effects. This suggests care must be taken in attributing deviations from vacuum general relativity to intrinsic quantum gravity signals when a matter halo is present. The paper's analytic hierarchy implies that shadow/ringdown/energy emission observables probe the ambient matter distribution much more robustly than nonlinear QED.

Theoretically, the study motivates extending this framework to axisymmetric spinning solutions, coupling to more realistic dark matter models, and considering higher-order QED corrections for extremal configurations. Further, the sparsity analysis accentuates the fundamentally quantum nature of black hole evaporation even with matter halos, reinforcing the need for quantum gravity completions in such backgrounds.

Conclusion

The analysis establishes a clear dominance of PFDM-induced effects over Euler–Heisenberg electrodynamics in shaping the optical, dynamical, and radiative properties of charged black holes in the leading-parameter regime. The PFDM parameter $R_{\rm sh}$3 sets the scale for shadow contraction, QNM oscillation and damping, grey-body spectral thresholds, and Hawking temperature enhancement, while the EH coupling $R_{\rm sh}$4 imprints only a perturbative correction that is detectable predominantly in high-charge situations. Consequently, black hole shadow and ringdown signals are more effective diagnostics of dark matter environments than of strong-field QED effects in this model. Future work should pursue axes of observational discrimination and higher-order corrections, with implications for both astrophysical black hole phenomenology and the program of quantum gravity phenomenology.

Reference: "Shadow, Quasinormal Modes, Sparsity, and Energy Emission Rate of Euler-Heisenberg Black Hole Surrounded by Perfect Fluid Dark Matter" (2604.16628).