Observational properties and quasinormal Modes of the Hayward black Hole surrounded by a cloud of strings

TL;DR

This work investigates a Hayward regular black hole embedded in a string cloud, focusing on how the regularization parameter $l$ and string-cloud parameter $a$ affect horizon structure, photon trajectories, shadow features, disk imaging, and quasinormal modes. The authors derive the metric function $f(r)$ and analyze geodesics to obtain $r_{-}$, $r_{+}$, $r_{ ext{ph}}$, $b_c$, and $r_{ ext{isco}}$, and use the Novikov-Thorne model to generate accretion-disk images, revealing that $l$ modestly modifies the geometry while $a$ shifts the disk structure and flux. For perturbations, scalar and electromagnetic fields are treated via Klein-Gordon and Maxwell equations, with time-domain evolution and sixth-order WKB rendering QNM frequencies; larger $l$ increases Re$\omega$ and tightens the potential barrier, whereas larger $a$ has the opposite effect and both reduce |Im$\omega$|, implying longer-lived oscillations. The results illuminate the coupling between regular black holes and surrounding string clouds and offer observable signatures, with M87* shadow constraints favoring small $a$ values and modest $l$-driven effects on imaging and damping rates.

Abstract

In this work, we explored the Hayward black hole surrounded by a cloud of strings, with a focus on the effects of the regularization parameter $l$ and the string cloud parameter $a$ on its observational properties and quasinormal modes (QNMs). Utilizing the spacetime metric and geodesic equations, we calculated several geometric quantities characterizing the black hole. To visualize the observational appearance of the accretion disk, we employed the Novikov--Thorne model to simulate both its primary and secondary images. Furthermore, we analyzed the QNMs of the black hole under scalar and electromagnetic perturbations for different parameter values. The results indicate that as the regularization parameter $l$ increases, the outer horizon radius $r_{+}$, photon-sphere radius $r_{\text{ph}}$, critical impact parameter $b_{c}$, and innermost stable circular orbit $r_{\text{isco}}$ exhibit a gradual decrease, while the inner horizon radius $r_{-}$ and the real part of the QNMs frequency $ω_{r}$ increase. In contrast, as the string cloud parameter $a$ increases, $r_{+}$, $r_{\text{ph}}$, $b_{c}$, and $r_{\text{ isco}}$ demonstrate a rapid increase, whereas $r_{-}$ and $ω_{r}$ decrease. In both cases, the absolute value of the imaginary part of the QNMs frequency decreases with the increase $l$ or $a$. This work offers a theoretical foundation for understanding the coupling between regular black holes and surrounding string clouds.

Figures (10)

• Figure 1: Variation curves of $f(r)$ with $r$ under different parameters, with $a = 0.1$ (left) and $l = 0.5$ (right)
• Figure 2: Deflection angle $\varphi(b)$ corresponding to intersections as a function of $b$ for different $r$, with $l=0.9,a=0.1$ (left); $l=0.9,a=0.2$ (right).
• Figure 3: Direct and indirect images of equal-r orbits of accretion disks under different parameters.
• Figure 4: Direct and indirect images of equal-r orbits of accretion disks under different parameters.
• Figure 5: Direct and indirect intensity images of accretion disks under different parameters.