Quasinormal modes of Schwarzschild-like black hole surrounded by the pseudo-isothermal dark matter halo

Abstract: The merger of binary black holes produces a series of decaying oscillations, during which energy is radiated in gravitational waves. The characteristic signal in the ringdown phase can be described by complex oscillation frequencies called quasinormal modes (QNMs). In this paper, we investigate the ringdown spectrum resulting from scalar field perturbations of black holes surrounded by pseudo-isothermal dark matter halos. The complex frequencies of these quasinormal modes are numerically computed using the sixth-order WKB approximation. Additionally, the time evolution of the scalar perturbations is examined using the finite difference method, considering various multipole numbers and dark matter halo parameters. For a static, spherically symmetric black hole, the photon sphere--composed of circular null geodesics--plays a crucial role in analyzing the black hole shadow. Furthermore, the connection between the black hole shadow and QNMs is explored in the eikonal limit.

• The paper presents a novel method to compute quasinormal mode frequencies in Schwarzschild-like black holes using both numerical and analytical techniques that incorporate dark matter halo parameters.
• It demonstrates how dark matter halo parameters, such as core radius and central density, significantly affect black hole shadows and the decay rates of scalar perturbations.
• The study establishes a connection between the angular velocity of photon orbits and quasinormal mode oscillations, offering new insights into constraining dark matter distributions via astronomical observations.

Quasinormal Modes of Schwarzschild-like Black Holes Surrounded by Pseudo-Isothermal Dark Matter Halo

The paper presents a comprehensive study of quasinormal modes (QNMs) of Schwarzschild-like black holes surrounded by pseudo-isothermal dark matter halos. It investigates scalar field perturbations using both numerical and analytical methods to evaluate the QNM frequencies, consider the time evolution of perturbations, and explore the connection between black hole shadows and QNMs. This contributes significantly to understanding the interaction between dark matter halos and black holes, along with the impacts on gravitational wave propagation.

Methodology

Schwarzschild-like Black Hole Model with Pseudo-Isothermal Halo

The authors derive the Schwarzschild-like metric for a black hole immersed in a pseudo-isothermal dark matter halo by solving the Einstein field equations. The metric function $f(r)$ includes terms accounting for dark matter halo parameters: the core radius $r_0$ and the central density $\rho_0$. This metric recovers to the classical Schwarzschild solution when $\rho_0 = 0$, indicating no dark matter presence.

The pseudo-isothermal halo assumes a density profile that remains constant at the center, with a gradual tapering, consistent with empirical observations of low surface brightness galaxies. This model helps avoid singularities inherent in other profiles, like the NFW profile.

Analysis of Black Hole Shadow

The authors evaluate the black hole shadow using the photon sphere radius derived from circular null geodesics. The shadow’s size corresponds to the photon sphere’s unstable circular orbits, calculated via numerical methods. The paper illustrates how both $r_0$ and $\rho_0$ influence the shadow's radius, showing significant variation as these parameters change, especially at high density values.

Computation of Quasinormal Modes

QNMs characterize the ringdown phase following a perturbation, represented by complex frequencies indicating both oscillation frequency and decay rate. The paper uses the sixth-order WKB approximation to compute these frequencies efficiently. The effective potential exhibits a potential barrier, and QNM frequencies depend significantly on dark matter parameters. The study confirms stable solutions for the black hole due to the potential barrier, verifying its oscillatory properties.

Eikonal Limit and Shadow-QNM Connection

A significant aspect discussed is the eikonal limit where the real and imaginary components of QNM frequencies relate to the shadow size. The angular velocity $\Omega_{r_{ph}}$ and the Lyapunov exponent $\lambda_L$ describe the behavior of orbits and instability. The correlation provides a pathway to deducing QNM characteristics from astronomical observations of shadows.

Numerical Simulation of Time Evolution

Utilizing a time domain integration method with finite differences, the paper demonstrates the evolution of scalar perturbations in the black hole’s vicinity. This reveals the impact of varying $r_0$ and $\rho_0$ on damping rates and oscillation frequencies in the temporal evolution of QNMs. Larger values of dark matter parameters accelerate system damping and enhance oscillatory behavior.

Implications and Future Directions

The research highlights that varying dark matter halo parameters substantively affect QNMs and gravitational wave propagation around black holes. This could improve constraints on halo characteristics via observational data, enhancing our understanding of dark matter distribution at galactic centers.

Future work might explore extending these findings to rotating Kerr black holes and investigate the influence of non-spherically symmetric halo distributions. Such studies could unravel more features of halo-RBH interactions, aiding in constructing robust models for galaxies hosting supermassive black holes.

Conclusion

The paper elucidates critical aspects of quasinormal modes in Schwarzschild-like black holes enveloped by pseudo-isothermal dark matter halos. Through detailed methodological approaches, it thoroughly details QNM behaviors and their implications, revealing intersections between shadow radii and QNMs in the eikonal limit. The findings significantly contribute to the field of black hole physics and dark matter studies, offering pathways for future explorations.