Equatorial periodic orbits and gravitational waveforms in Bardeen black holes surrounded by perfect fluid dark matter

Technical Summary: Equatorial Periodic Orbits and Gravitational Waveforms in Bardeen Black Holes Surrounded by Perfect Fluid Dark Matter

Problem Statement
While General Relativity (GR) has been remarkably accurate in describing astrophysical phenomena, it suffers from the inevitability of singularities during gravitational collapse. Regular Black Holes (RBHs), such as the Bardeen black hole, offer a singularity-free alternative, often arising from specific forms of Non-Linear Electrodynamics (NED). Concurrently, the astrophysical environment of black holes is rarely a vacuum; it is often permeated by Dark Matter (DM). The interplay between the non-linear electromagnetic fields generating the Bardeen metric and the presence of DM halos remains an under-explored avenue for probing fundamental physics. This study addresses the need to understand how a Bardeen black hole, modified by a Perfect Fluid Dark Matter (PFDM) background, influences particle dynamics and the resulting gravitational wave signatures.

Methodology
The authors construct a static, spherically symmetric metric for a Bardeen black hole surrounded by PFDM (BPFDM BH) by minimally coupling NED and the DM matter field. The study focuses on equatorial plane motion of test particles.

1. Effective Potential Analysis: The authors derive the effective potential ($V_{eff}$) governing timelike geodesics. By imposing conditions on $V_{eff}$ and its derivatives, they analytically and numerically determine the locations and conserved quantities (energy $E$ and angular momentum $L$) for the Innermost Stable Circular Orbit (ISCO) and the Marginally Bound Orbit (MBO).
2. Periodic Orbit Taxonomy: The study investigates periodic orbits characterized by a rational number $q = \omega_\phi / \omega_r - 1$. These orbits are systematically organized using the $(z, w, v)$ taxonomy, where $z$ is the zoom number, $w$ is the whirl number, and $v$ is the vertex number. The authors analyze how $q$ varies with energy and angular momentum across different model parameters.
3. Gravitational Waveform Generation: To simulate Extreme Mass-Ratio Inspirals (EMRIs) where a stellar-mass object orbits a supermassive BPFDM BH, the authors employ the numerical "Kludge" method. This involves:
   • Solving geodesic equations to obtain particle trajectories in Boyer-Lindquist coordinates.
   • Projecting these trajectories onto a Cartesian frame.
   • Applying the quadrupole formula in the weak-field limit to generate gravitational waveforms ($h_+$ and $h_\times$).
   • Analyzing the waveforms in the context of zoom-whirl motion.

Key Contributions and Results

• Metric and Regularity: The derived metric for the BPFDM BH reduces to the standard Bardeen metric when the DM parameter $\beta \to 0$, and to the Schwarzschild metric when both $\beta \to 0$ and the magnetic charge $g \to 0$. The authors note that the presence of PFDM renders the black hole no longer "regular" in the strict sense, as the DM background introduces new features.
• Orbital Dynamics (ISCO and MBO):
  • Both the magnetic charge ($g$) and the DM parameter ($\beta$) significantly alter the orbital dynamics.
  • Increasing $g$ or $\beta$ causes the radii of the ISCO ($r_{ISCO}$) and MBO ($r_{MBO}$) to decrease.
  • The angular momentum ($L_{ISCO}$, $L_{MBO}$) required for these orbits decreases with increasing $g$ and $\beta$.
  • The energy ($E_{ISCO}$) decreases with increasing $g$ but increases with increasing $\beta$.
  • The study finds that the DM parameter has a steeper impact on ISCO-related quantities than the magnetic charge.
• Periodic Orbits and the $(z, w, v)$ Taxonomy:
  • The rational number $q$ diverges as angular momentum approaches its lower bound, signaling the onset of extreme "whirl" motion.
  • Higher values of $g$ or $\beta$ shift the onset of this extreme whirling motion to lower angular momenta and higher energies.
  • For a fixed $(z, w, v)$ configuration, the required energy and angular momentum decrease as either $g$ or $\beta$ increases.
  • Increasing the whirl number $w$ keeps the particle in the strong-field region near the horizon for a longer duration, while increasing the zoom number $z$ increases the number of elliptical excursions.
• Gravitational Waveforms:
  • The generated waveforms exhibit distinct "zoom-whirl" signatures. The "zoom" phase (weak field) corresponds to low-amplitude, less oscillatory segments, while the "whirl" phase (strong field) corresponds to high-amplitude, high-frequency peaks.
  • Impact of Parameters: Increasing either $g$ or $\beta$ amplifies the gravitational wave amplitude and decreases the orbital time period (phase shift).
  • Comparative Impact: The study observes that while both parameters enhance gravitational radiation, the effect of the DM parameter ($\beta$) is significantly more conspicuous than that of the magnetic charge ($g$).

Significance and Claims
The paper claims that the combined effects of Non-Linear Electrodynamics and Perfect Fluid Dark Matter leave distinct, observable imprints on both orbital dynamics and gravitational waveforms. Specifically, the presence of DM and magnetic charge modifies the effective potential, leading to tighter bound orbits and altered transition states in EMRIs. The authors posit that these distinct signatures—particularly the pronounced effect of the DM parameter on waveform amplitude and orbital timescales—could serve as a probe for the nature of the central supermassive black hole and its surrounding environment. The study suggests that future observations of gravitational waves from EMRIs could test the efficacy of this model and potentially distinguish between vacuum black holes and those embedded in DM halos with NED origins.