Equatorial periodic orbits and gravitational wave signatures in Euler-Heisenberg black holes surrounded by perfect fluid dark matter

Abstract: We investigate equatorial periodic orbits and their gravitational wave radiation in the spacetime of an Euler--Heisenberg (EH) black hole surrounded by perfect fluid dark matter (PFDM). The combined effects of quantum electrodynamic corrections and dark matter are incorporated through an effective metric, and the dynamics of timelike geodesics are analyzed using the effective potential formalism. We derive the conditions for marginally bound and innermost stable circular orbits, classify periodic trajectories using the rational parameter and topological indices, and identify a rich hierarchy of zoom--whirl motions in the strong-field regime. Gravitational wave signals from periodic orbits are computed using the numerical kludge method, revealing characteristic burst-like features associated with whirl phases. Our results show that perfect fluid dark matter systematically modifies the stability thresholds and suppresses the waveform amplitude, while QED corrections enhance high-frequency components generated near the horizon. These findings demonstrate that periodic orbits in the EH--PFDM spacetime provide a sensitive probe of quantum corrections and dark matter effects in strong gravitational fields.

• The paper demonstrates how quantum electrodynamical corrections and PFDM modify equatorial periodic orbits, altering ISCO, MBO, and zoom–whirl dynamics.
• It employs the effective potential formalism and numerical kludge methods to derive gravitational waveform templates for extreme mass-ratio inspirals.
• Results show that increased dark matter density dilutes gravitational attraction, extending orbits and suppressing waveform amplitudes.

Equatorial Periodic Orbits and Gravitational Wave Signatures in Euler–Heisenberg Black Holes Surrounded by Perfect Fluid Dark Matter

Introduction

This paper presents an in-depth analysis of timelike equatorial periodic orbits and their gravitational wave (GW) emission in the Euler–Heisenberg (EH) black hole spacetime with an external perfect fluid dark matter (PFDM) distribution (2604.11866). The work systematically examines how quantum electrodynamical (QED) corrections to classical electrovacuum gravity, as encapsulated by the EH action, and large-scale environmental effects due to PFDM alter strong-field orbital dynamics and associated GW phenomenology, with explicit focus on innermost stable circular orbits (ISCOs), marginally bound orbits (MBOs), and highly precessing "zoom–whirl" trajectories. The study employs the effective potential formalism for geodesics, provides a topological taxonomy of periodic orbits, and generates waveform templates using the numerical kludge approximation appropriate for extreme mass-ratio inspirals (EMRIs).

Euler–Heisenberg–PFDM Black Hole Metric

The adopted spacetime metric follows from the Einstein-Hilbert action augmented with the leading-order EH nonlinear electromagnetic Lagrangian and a PFDM energy-momentum tensor. The line element is

$$ds^2 = -g(r)\,dt^2 + \frac{dr^2}{g(r)} + r^2 (d\theta^2 + \sin^2\theta\,d\phi^2)$$

with

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

Here, $M$, $Q$, $a$, and $\alpha$ are, respectively, the ADM mass, black hole electric charge, QED interaction parameter, and the PFDM environment parameter.

The QED correction $a$ is subdominant at large $r$ but crucial for near-horizon physics; the PFDM term, characterized by $\alpha$, dominates at large and intermediate radii, modifying the global gravitational potential. The limits $a\to 0$, $$g(r) = 1 - \frac{2M}{r} + \frac{Q^2}{r^2} - \frac{aQ^4}{20r^6} + \frac{\alpha}{r}\ln\left(\frac{r}{|\alpha|}\right).$$0, $$g(r) = 1 - \frac{2M}{r} + \frac{Q^2}{r^2} - \frac{aQ^4}{20r^6} + \frac{\alpha}{r}\ln\left(\frac{r}{|\alpha|}\right).$$1 recover, in turn, the PFDM-Schwarzschild, EH–Reissner–Nordström, or classical Reissner–Nordström solutions.

Effective Potential and Strong-Field Orbital Structure

The motion of test particles is governed by the effective potential

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

for particles with angular momentum $$g(r) = 1 - \frac{2M}{r} + \frac{Q^2}{r^2} - \frac{aQ^4}{20r^6} + \frac{\alpha}{r}\ln\left(\frac{r}{|\alpha|}\right).$$3. The orbital dynamics are reviewed via this effective potential framework, with the usual characterization of circular, bound, plunging, and marginal orbits.

Figure 1: Effective potential $$g(r) = 1 - \frac{2M}{r} + \frac{Q^2}{r^2} - \frac{aQ^4}{20r^6} + \frac{\alpha}{r}\ln\left(\frac{r}{|\alpha|}\right).$$4 for the EH–PFDM black hole across different $$g(r) = 1 - \frac{2M}{r} + \frac{Q^2}{r^2} - \frac{aQ^4}{20r^6} + \frac{\alpha}{r}\ln\left(\frac{r}{|\alpha|}\right).$$5, revealing the outward shift and shallowing of the potential well with increasing PFDM content.

The PFDM parameter $$g(r) = 1 - \frac{2M}{r} + \frac{Q^2}{r^2} - \frac{aQ^4}{20r^6} + \frac{\alpha}{r}\ln\left(\frac{r}{|\alpha|}\right).$$6 modulates both the height of the barrier (associated with the unstable circular orbit) and the depth of the bound region. Increasing $$g(r) = 1 - \frac{2M}{r} + \frac{Q^2}{r^2} - \frac{aQ^4}{20r^6} + \frac{\alpha}{r}\ln\left(\frac{r}{|\alpha|}\right).$$7 systematically shifts the ISCO and MBO radii outward, decreases $$g(r) = 1 - \frac{2M}{r} + \frac{Q^2}{r^2} - \frac{aQ^4}{20r^6} + \frac{\alpha}{r}\ln\left(\frac{r}{|\alpha|}\right).$$8 and $$g(r) = 1 - \frac{2M}{r} + \frac{Q^2}{r^2} - \frac{aQ^4}{20r^6} + \frac{\alpha}{r}\ln\left(\frac{r}{|\alpha|}\right).$$9, and reduces the permitted domain for stable bound orbits. This is consistent with PFDM's effective dilution of gravitational attraction and potential flattening at large radii.

Figure 2: Dependence of ISCO radius $M$0, angular momentum, and energy on $M$1; all orbital thresholds shift toward larger radii with increasing PFDM density.

The critical loci in the $M$2–$M$3 parameter space clarify the shrinking of the bound region and the enhanced propensity for plunging/escaping trajectories when PFDM is present.

Figure 3: Allowed parameter regions for bound timelike orbits as a function of $M$4, showing the shrinking of accessible phase space for periodic orbits with increased dark matter.

Periodic Orbits and Zoom–Whirl Hierarchy

The paper adopts the standard rational frequency parameterization for periodic orbits and classifies relativistic bound motion using the $M$5 indices, corresponding to zooms, whirls, and vertices (precessions), respectively. The rational parameter

$M$6

is computed for families of orbits, providing a granular mapping between energy, angular momentum, and zoom–whirl behavior.

Figure 4: Rational parameter $M$7 versus particle energy for fixed $M$8 in the EHPFDM black hole.

The divergence of $M$9 as energy approaches the critical value for the unstable circular orbit signals the transition to extreme zoom–whirliness.

Figure 5: Rational parameter $Q$0 as a function of $Q$1 for given energy; increasing $Q$2 shifts divergence to lower $Q$3, facilitating zoom–whirl orbits for a wider parameter subset.

Both energy and angular momentum regulate the nature and extent of zoom–whirl morphologies.

Figure 6: Representative periodic orbits for fixed energy and various $Q$4 values, highlighting zoom and whirl variations.

Figure 7: Periodic orbits for fixed $Q$5 and different $Q$6, demonstrating energy's effect on excursion size and angular momentum's effect on whirl depth.

The topological structure encoded by these indices provides fine diagnostic power—PFDM and QED corrections modulate both the global size and strong-field content of orbits, reshaping the spectrum of periodic motion.

Gravitational Waveforms from Strong-Field Periodic Orbits

Utilizing the numerical kludge approach, the work generates GW polarizations $Q$7 and $Q$8 for fiducial EMRIs (e.g., $Q$9 secondary, $a$0 primary). The GW signal exhibits the expected correspondence between zoom and whirl phases—the former produces broad, lower-amplitude modulations, while the latter generates localized, high-frequency, high-amplitude bursts.

Figure 8: Gravitational waveform for $a$1; left shows the associated zoom–whirl orbit, right the characteristic GW polarizations with bursty structure during the whirl phase.

Systematic exploration of parameter dependence yields the following empirical results:

• Increasing PFDM $a$2: Orbital geometry becomes more extended, periapsis location increases, and GW amplitudes are suppressed due to the reduced time spent in the strong-field region and weaker accelerations.

  Figure 9: Increasing $a$3 for $a$4 deforms the orbit and suppresses both amplitude and high-frequency content of the waveform.

• Varying electric charge $a$5: Larger $a$6 amplifies whirl content in periodic orbits and enhances the high-frequency GW components due to strengthened near-horizon curvature.

  Figure 10: Increasing $a$7 intensifies whirl behavior and high-frequency GW components.

• QED parameter $a$8: Modifies near-horizon orbital structure, giving moderate but systematic changes in waveform amplitude and instantaneous frequency, reflecting the sensitive influence of vacuum polarization corrections in the strong-field regime.

  Figure 11: Increasing $a$9 for fixed $\alpha$0 and $\alpha$1 alters the strong-field segment of the orbit, resulting in amplitude and frequency modifications in the waveform.

The waveform morphology reflects a direct imprint of the relativistic zoom–whirl structure and the microphysical (QED) and macrophysical (PFDM) background modifications.

Implications and Outlook

The analysis strengthens the conclusion that relativistic periodic orbits are highly sensitive to both environmental and quantum modifications of the black hole metric. Since the GW signals encode the orbital precession, zoom–whirl content, and near-horizon geometry, they serve as robust probes of underlying microphysics and dark matter distributions.

Practically, these results inform template construction for future GW searches targeting non-vacuum black hole environments, particularly in the context of LISA/space-based detectors where EMRI signals predominate. Theoretically, they show that deviations from general relativistic predictions (e.g., ISCO radius, precession rate, waveform morphology) could be diagnostic for new physics such as strong-field QED effects or non-baryonic matter distributions.

Future work should extend to dissipative phenomena (i.e., full radiation reaction and self-force), rotate the background, and perform a multi-messenger synthesis incorporating shadow, lensing, and QPO observables for the EH–PFDM metric class.

Conclusion

The composite analysis of geodesic structure and waveform emission for test particles in EH–PFDM backgrounds reveals that periodic orbit spectroscopies and associated GWs are sensitive tools for probing QED-induced and dark matter-induced deviations from the standard black hole paradigm. PFDM suppresses and smooths GW amplitudes by diluting the central potential, while QED and charge enhance strong-field dynamical features. These findings motivate the use of periodic orbit frameworks in future theoretical and data analysis pipelines for strong-field gravity and astrophysical black hole studies.