Quasinormal modes of massless scalar and electromagnetic perturbations for Euler Heisenberg black holes surrounded by perfect fluid dark matter

Abstract: We investigate the quasinormal modes of massless scalar and electromagnetic perturbations in charged Euler--Heisenberg black holes surrounded by perfect fluid dark matter. The quasinormal frequencies are calculated using the asymptotic iteration method and the sixth-order WKB approximation, and the relative deviation between the two methods is quantitatively analyzed to verify the reliability of results. The greybody factors for both perturbations are also evaluated within the sixth-order WKB framework. We systematically examine the effects of the black hole charge $Q$, nonlinear electrodynamic parameter $a$, dark matter parameter $λ$, and angular quantum number $l$ on the quasinormal frequencies and greybody factors. We find that these parameters significantly modify the structure of the effective potential barriers, and thus affect the oscillation frequencies, damping rates, and wave transmission and reflection properties of the perturbed fields.

• The paper computes fundamental scalar and electromagnetic quasinormal modes using asymptotic iteration and sixth-order WKB methods, achieving agreement better than 10⁻³% for l≥2 and below 0.08% for l=1.
• The analysis finds that dark matter produces nonmonotonic spectral changes, while sufficiently strong dark matter removes the charge-induced minimum in damping and increases the importance of the background environment over Euler–Heisenberg corrections.
• The paper shows that charge and dark matter suppress low-frequency greybody transmission, whereas high-frequency transmission approaches unity, providing theoretical templates for black-hole spectroscopy in dark-matter halos.

Background and motivation

The ringdown phase of a perturbed black hole is governed by quasinormal modes (QNMs), whose complex frequencies encode both the parameters of the background geometry and possible deviations from general relativity. Two physical ingredients motivate the spacetime studied in this paper: nonlinear electrodynamics, via the Euler–Heisenberg (EH) effective action that captures one-loop QED vacuum polarization corrections to Maxwell theory, and perfect fluid dark matter (PFDM), a phenomenological model in which the dark matter halo introduces logarithmic corrections to the metric function. The paper computes QNMs of massless scalar and electromagnetic perturbations on the asymptotically flat EH black hole surrounded by PFDM, together with greybody factors, using two independent numerical schemes.

Spacetime and effective potentials

The static, spherically symmetric metric has

$$f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}-\frac{aQ^4}{20r^6}+\frac{\lambda}{r}\ln\left|\frac{r}{\lambda}\right|,$$

where $Q$ is the magnetic charge, $a$ the EH parameter, and $\lambda$ the PFDM intensity. The solution interpolates between known limits: $\lambda\to0$ gives the EH black hole, $Q\to0$ gives Schwarzschild-PFDM, and both vanishings recover Reissner–Nordström or Schwarzschild. The horizon analysis shows that PFDM reduces the extremal charge $Q_c$, positive $a$ increases it, and negative $a$ decreases it; for negative $a$ below a critical value $Q$0 a naked singularity forms.

Scalar perturbations obey a Schrödinger-like equation with potential $Q$1, while axial and polar electromagnetic sectors unify into a single equation with $Q$2. The key structural difference is the derivative term $Q$3 present only in the scalar potential; this term is responsible for most of the spin-dependent behavior found later. Both potentials rise with $Q$4 and $Q$5, respond weakly to moderate $Q$6, and depend nonmonotonically on $Q$7: small-to-moderate $Q$8 raises and sharpens the barrier, while large $Q$9 lowers and broadens it.

Numerical methods and validation

Fundamental ($a$0) frequencies are computed with the asymptotic iteration method (AIM), implemented with a compactified coordinate $a$1 and an ansatz enforcing ingoing boundary conditions at the horizon and outgoing conditions at infinity, and independently with the sixth-order WKB approximation of Konoplya. Agreement is quantified by the relative deviation $a$2. For $a$3 the deviation is at the level of $a$4 or better; for $a$5 it stays below roughly $a$6. Only the monopole scalar mode ($a$7) shows deviations near $a$8, which is expected since WKB converges poorly when the barrier is shallow. This pattern validates the spectrum everywhere except possibly the lowest multipole, where WKB accuracy is intrinsically limited.

Dependence of the quasinormal spectrum

Angular momentum: increasing $a$9 raises $\lambda$0 monotonically for both fields, as expected from the higher barrier. The damping behaves oppositely for the two spins: $\lambda$1 decreases with $\lambda$2 for scalars but increases for electromagnetics, a distinction traced to the different near-peak curvature dependence of $\lambda$3 versus $\lambda$4. Dark matter amplifies both $\lambda$5 and $\lambda$6, more so at larger $\lambda$7.

Dark matter parameter: the $\lambda$8-dependence is nonmonotonic for both fields — $\lambda$9 peaks and $\lambda\to0$0 dips at critical values of $\lambda\to0$1 (e.g., for scalar $\lambda\to0$2 the maxima lie near $\lambda\to0$3). Notably, the critical $\lambda\to0$4 decreases with $\lambda\to0$5 for scalars but increases with $\lambda\to0$6 for electromagnetics, again reflecting the extra derivative term in $\lambda\to0$7. Increasing $\lambda\to0$8 shifts these critical values downward, and increasing $\lambda\to0$9 moves the maximum-$Q\to0$0 point to larger $Q\to0$1 while moving the minimum-damping point to smaller $Q\to0$2; in the large-$Q\to0$3 regime the influence of $Q\to0$4 fades and PFDM dominates the dynamics.

Nonlinear parameter: for moderate $Q\to0$5, both $Q\to0$6 and $Q\to0$7 vary monotonically with $Q\to0$8, with electromagnetic modes less sensitive than scalar ones. Larger $Q\to0$9 suppresses the effect of $Q_c$0, while larger $Q_c$1 and the presence of PFDM amplify it. In the strong-nonlinear regime the differences between endpoints of the $Q_c$2-scan themselves peak at a critical $Q_c$3, mirroring the nonmonotonicity seen elsewhere.

Charge: $Q_c$4 grows monotonically with $Q_c$5 for moderate $Q_c$6, driven by the rising barrier height. The damping exhibits a robust nonmonotonic structure, with $Q_c$7 reaching a minimum near $Q_c$8–$Q_c$9 before growing toward extremality. A striking result is that for $a$0 this nonmonotonic damping disappears entirely and $a$1 becomes monotonic in $a$2 — sufficiently strong dark matter qualitatively erases the charge-induced damping signature. For large $a$3, the monotonic growth of $a$4 with $a$5 degrades into a rise-then-fall trend near extremality. Throughout, the extremal charge $a$6 increases monotonically with $a$7, confirming that stronger EH corrections push the extremal condition to larger charge.

Greybody factors

Using sixth-order WKB transmission coefficients, $a$8, the paper finds strong low-frequency suppression and $a$9 at high frequency, as dictated by the barrier. Increasing $a$0 suppresses low-frequency transmission and shifts the rising edge upward, an effect strengthened by PFDM. Variations of $a$1 produce only weak modifications, confined mostly to the low-frequency region where negative $a$2 slightly suppresses transmission. Moderate $a$3 raises and narrows the barrier, further suppressing tunneling. Scalar and electromagnetic greybody factors behave qualitatively identically, indicating that scattering is governed primarily by the background geometry rather than field spin.

Limitations and open questions

Several caveats should be noted. First, all results are restricted to fundamental modes; overtones are not computed, although they are precisely the sector where WKB-based methods are least reliable and where dark matter effects could differ. Second, the WKB treatment of the $a$4 scalar mode carries percent-level uncertainty, so quantitative claims about the monopole rest mainly on AIM alone. Third, the analysis treats the fields as test perturbations on a fixed background and considers only massless fields; gravitational perturbations, which dominate observable ringdowns, are not addressed. Fourth, the PFDM model is phenomenological, and mapping the parameter ranges explored here onto realistic halo densities around astrophysical black holes remains unquantified. Finally, whether the nonmonotonic damping signatures reported here survive in the full gravitational sector, or are distinguishable against astrophysical noise, is left open.

Conclusion

This work provides a systematic, finite-$a$5 quasinormal spectrum and greybody-factor analysis for charged Euler–Heisenberg black holes embedded in perfect fluid dark matter, cross-validated by AIM and sixth-order WKB with sub-percent agreement for $a$6. The principal findings are the nonmonotonic dependence of the spectrum on $a$7, the charge-induced minimum in damping that strong dark matter can eliminate entirely, the monotonic increase of the extremal charge with the EH parameter, and the consistently dominant roles of $a$8 and $a$9 relative to the mild near-horizon corrections from $a$0. These results supply theoretical templates for black-hole spectroscopy in dark matter environments, though extension to gravitational perturbations and overtones remains necessary for direct observational application.