Ringdown of a black hole embedded in a Burkert dark matter halo

TL;DR

This work addresses how a cored Burkert dark matter halo backreacts on a Schwarzschild black hole and modifies its ringdown signature. It constructs an analytic BH metric embedded in a Burkert halo by fixing the halo geometry from rotation curves and solving the Einstein equations with Burkert stress-energy under Schwarzschild boundary conditions, yielding a deformation $F(r)$ that recovers Schwarzschild in the halo-less limit. Linear perturbations of spins $s=0,1,2$ are analyzed using Leaver’s continued fraction method, high-order WKB, and time-domain evolutions, revealing that larger halo core radius $r_0$ or central density $\rho_0$ generally increase both the oscillation frequency $\Re(\omega)$ and damping rate $|\Im(\omega)|$, while larger multipole $l$ mainly raises $\Re(\omega)$. The two frequency extraction methods agree within small systematic offsets, validating the results and providing a benchmark for gravitational-wave tests of DM halos. The findings offer a controlled setting to assess environmental effects on BH ringdowns and motivate extensions to rotating BHs and other halo models, with potential implications for multi-messenger probes of DM distributions around BHs.

Abstract

We construct a new static, spherically symmetric black hole spacetime embedded in a dark matter halo whose density follows the cored Burkert profile. Starting from the halo-only geometry determined by the rotation curve relation, we solve the Einstein equations with the Burkert stress-energy and enforce a Schwarzschild boundary condition, obtaining closed form metric functions in which the halo contribution deforms the redshift or shape functions and reduces to the Schwarzschild limit when the halo parameters vanish. On this background we study linear perturbations of test fields with spins $s=0,1,2$ and compute their quasinormal spectra using both a high order WKB scheme and continued fraction method, complemented by time domain evolutions. We find that increasing either the Burkert core radius $r_0$ or the central density $ρ_0$ generically shifts the real part of the frequencies upward and enhances damping, while the multipole index $l$ primarily increases the oscillation frequency with a milder impact on the decay rate. The two frequency extraction methods agree to within small, systematic offsets across the explored parameter space. Our results quantify how a cored dark matter environment imprints itself on the ringdown of a central black hole and provide benchmarks for future gravitational wave tests of halo properties.

Figures (6)

• Figure 1: The plot illustrates the relationship between the real and imaginary parts of the QNM frequencies using the WKB method for different values of $\rho_0$ from Table I.
• Figure 2: The plot above illustrate the relationship between the real and imaginary parts of the QNM frequencies using the WKB method for $r_0$ from Table I.
• Figure 3: The plot above illustrate the relationship between the real and imaginary parts of the QNM frequencies using the WKB method for $l$ from Table I.
• Figure 4: The time-domain profiles of the scalar field perturbation for different $r_0$ with $M=0.5,l=1,\rho_0=0.1$, for different $\rho_0$ with $M=0.5,l=1,r_0=0.1$, and for different $l$ with $M=0.5,r_0=0.1,\rho_0=0.1$, respectively.
• Figure 5: The time-domain profiles of the electromagnetic field perturbation for different $r_0$ with $M=0.5,l=1,\rho_0=0.1$, for different $\rho_0$ with $M=0.5,l=1,r_0=0.1$, and for different $l$ with $M=0.5,r_0=0.1,\rho_0=0.1$, respectively.