Probing the Charged Hayward Black Hole in Dark Matter and String Cloud Environments through Shadow, Geodesics, and Quasinormal Spectrum

TL;DR

We study a regular charged Hayward black hole embedded in a cloud of strings and surrounded by perfect fluid dark matter, described by the metric function $f(r)=1-\alpha-\frac{2 M r^2}{r^3+g^3}+\frac{Q^2}{r^2}+\frac{\beta}{r}\ln\frac{r}{|\beta|}$. Through analytic and numerical methods, we analyze horizon structure, null and timelike geodesics, QPOs, scalar perturbations, and greybody factors, highlighting how the CoS parameter $\alpha$ and PFDM parameter $\beta$ imprint distinct observational signatures, while the Hayward parameter $g$ regularizes the core and the electric charge $Q$ modifies the EM contribution. We show that $\alpha$ and $\beta$ generally enlarge the photon sphere and shadow, derive the shadow radius $R_{\rm sh}$ for both distant and finite observers, and establish a link between shadow size and QNM frequencies in the eikonal limit. The results suggest that independent constraints on $α$ and $β$ are possible via shadow measurements (EHT), QPO timing, and gravitational-wave ringdown observations, offering a multi-messenger probe of regular BHs in realistic astrophysical environments.

Abstract

We construct a charged Bardeen black hole (BH) surrounded by perfect fluid dark matter (PFDM) and coupled to a cloud of strings (CS). The metric function combines the magnetic monopole charge from nonlinear electrodynamics, the PFDM logarithmic correction, and the CS parameter that renders the spacetime asymptotically non-flat. We analyze the horizon structure, identifying parameter ranges yielding non-extremal BHs, extremal configurations, and naked singularities. The null geodesics, photon sphere radius, and shadow are computed, revealing that both CS and PFDM enlarge the shadow. For neutral particle dynamics, we derive the specific energy, angular momentum, and innermost stable circular orbit location. Quasiperiodic oscillations (QPOs) are examined through the azimuthal, radial, and vertical epicyclic frequencies, where notably the azimuthal frequency is independent of the CS parameter. Scalar field perturbations governed by the Klein-Gordon equation yield an effective potential whose peak decreases with both parameters, yet the transmission and reflection probabilities respond oppositely to CS and PFDM variations. The greybody factor bounds are obtained using semi-analytical methods. Our results demonstrate that the distinct effects of $α$ and $β$ on various observables could allow independent constraints on these parameters through shadow measurements, QPO timing, and gravitational wave ringdown observations.

Figures (17)

• Figure 1: Metric function $f(r)$ versus $r/M$ for the charged Hayward BH with CoS and PFDM. The curves illustrate distinct horizon configurations: Schwarzschild (black solid, $r_h=2M$) serves as the reference; Hayward (dark green dashed, $g/M=0.3$) shows the regular BH with two horizons; Non-extremal (blue dash-dotted, $\alpha=0$, $\beta/M=0.8$) crosses $f(r)=0$ at two distinct radii $r_-$ and $r_+$; Extremal cases (red and purple solid curves) touch $f(r)=0$ tangentially at $r_{\rm ext} \approx 2M$, with filled circles marking the degenerate horizons; Single-horizon (magenta long-dashed, $\alpha=0.2$) exhibits one crossing; Naked singularity (orange dotted, $g/M=0.8$, $Q/M=1.2$) remains positive for all $r>0$. The gray horizontal line marks $f(r)=0$. Note that curves with $\alpha>0$ asymptote to $f(\infty) = 1-\alpha < 1$, reflecting the non-flat nature of CoS spacetimes.
• Figure 2: Three-dimensional visualization of the metric function $f(r)$ for the charged Hayward BH with CoS and PFDM. The panels correspond to: (a)$\alpha=0.05$, $\beta/M=0.3$, $g/M=0.2$, $Q/M=0.3$; (b)$\alpha=0.1$, $\beta/M=0.8$, $g/M=0.25$, $Q/M=0.6$; (c)$\alpha=0.1$, $\beta/M=0.72$, $g/M=0.25$, $Q/M=0.68$; (d)$\alpha=0.15$, $\beta/M=1.2$, $g/M=0.3$, $Q/M=0.8$. The event horizon is located where the surface intersects the $z=0$ plane. The blue spiral curves depict representative infalling trajectories.
• Figure 3: Behavior of the effective potential governing photon dynamics as a function of dimensionless radial distance $r/M$ for various $\beta/M$ and $\alpha$. Here $Q/M=1,\,g/M=0.25,\,\mathrm{L}/M=1$.
• Figure 4: Photon sphere radius as a function of $\alpha$ and $\beta/M$ for two values of the Hayward parameter $g$. Here $Q/M=1$.
• Figure 5: Shadow radius as a function of $\alpha$ and $\beta/M$ for two values of the Hayward parameter $g$. Here $Q/M=1$.