Observational signatures of charged Bardeen black holes in perfect fluid dark matter with a cloud of strings

TL;DR

This work investigates a regular charged Bardeen black hole embedded in perfect fluid dark matter (PFDM) and threaded by a cloud of strings (CS), described by a metric function $f(r)=1-\alpha-\frac{2Mr^2}{(q^2+r^2)^{3/2}}+\frac{Q^2}{r^2}+\frac{\beta}{r}\ln\left(\frac{r}{|\beta|}\right)$ that encodes the Bardeen magnetic charge, PFDM corrections, and CS deficit. The authors analyze horizon structure, null and timelike geodesics, photon sphere, black hole shadow, QPOs, scalar perturbations, and greybody factors, revealing that the CS parameter $\alpha$ and PFDM parameter $\beta$ have both cooperative and competing effects on observables. Notably, $\Omega_{\phi}$ is independent of $\alpha$, while $\Omega_r$ and $\Omega_\theta$ respond oppositely to $\alpha$ and $\beta$, offering multi-channel pathways to disentangle these contributions. The results suggest that combined shadow measurements, QPO timing, and gravitational-wave ringdown could independently constrain $\alpha$ and $\beta$, providing a multi-messenger probe of PFDM environments and string-cloud effects around black holes.

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 (16)

• Figure 1: Metric function $f(r)$ versus $r/M$ for the charged Bardeen BH with PFDM and CS. Six representative configurations are shown: Schwarzschild reference with $r_h = 2M$ (black solid); non-extremal BH with $(\alpha, \beta/M, q/M, Q/M) = (0, 1.0, 0.1, 1.0)$ having horizons at $r_h/M = [0.4559, 0.9679]$ (blue dashed); non-extremal BH with $(0.15, 1.2, 0.1, 1.0)$ and $r_h/M = [0.3314, 1.3105]$ (green dash-dotted); non-extremal BH with $(0, 0.5, 0.1, 0.5)$ and $r_h/M = [0.1547, 1.3102]$ (red dotted); single-horizon configuration with $(0.25, 1.8, 0.1, 1.0)$ showing $r_h/M = [0.1888, 1.8598]$ (magenta solid); and naked singularity with $(0, 0.5, 0.8, 1.5)$ where $f(r) > 0$ for all $r > 0$ (orange dashed). The horizontal gray line marks $f(r) = 0$, and intersections indicate horizon locations.
• Figure 2: Effective potential $M^2 V_{\rm eff}$ governing photon dynamics as a function of $r/M$ for various values of $\alpha$ and $\beta/M$. Panel (a): varying $\beta/M = 1.0, 1.2, 1.4, 1.6, 1.8, 2.0$ with fixed $\alpha = 0.1$. Panel (b): varying $\alpha = 0.05, 0.1, 0.15, 0.2, 0.25$ with fixed $\beta/M = 1.5$. Other parameters: $Q/M=1$, $q/M=0.1$. Increasing both $\beta/M$ and $\alpha$ suppresses the potential peak, reducing the gravitational barrier for photons.
• Figure 3: Effective radial force $M\,F_{\rm eff}$ on photons as a function of $r/M$. Panel (a): varying $\beta/M = 1.0, 1.2, 1.4, 1.6, 1.8$ with fixed $\alpha = 0.1$. Panel (b): varying $\alpha = 0.05, 0.1, 0.15, 0.2, 0.25$ with fixed $\beta/M = 0.8$. Other parameters: $Q/M=1$, $q/M=0.1$, $\mathrm{L}/M=1$. The force vanishes at the PS radius and approaches zero as $r \to \infty$.
• Figure 4: Three-dimensional surface plot of the PS radius $r_{\rm ph}/M$ as a function of $\alpha \in [0.05, 0.25]$ and $\beta/M \in [1.0, 1.8]$ with fixed $Q/M=1$, $q/M=0.1$. The PS radius increases monotonically with both parameters.
• Figure 5: Three-dimensional surface plot of the shadow radius $R_{\rm sh}/M$ as a function of $\alpha \in [0.05, 0.3]$ and $\beta/M \in [1.0, 1.8]$ with fixed $Q/M=1$, $q/M=0.3$. The shadow radius increases monotonically with both $\alpha$ and $\beta/M$.