Long-lived quasinormal frequencies for regular black hole supported by the Einasto profile in the presence of the magnetic field

Long-Lived Quasinormal Modes for Regular Black Holes with Einasto Halo and Magnetic Field

Introduction and Theoretical Context

This study analyzes the linear scalar perturbations of static, spherically symmetric regular black holes supported by the Einasto density profile in the presence of an external magnetic field, focusing on quasinormal mode (QNM) spectra, grey-body factors, and absorption cross-sections. The Einasto profile is an astrophysically motivated parameterization of dark matter halos and is widely supported by both high-resolution cosmological $N$-body simulations and empirical fits to observed galactic systems. Regular black holes with a nonsingular core, especially those constructed with physically motivated matter profiles such as Einasto, address the pathologies of classical singularities in the Schwarzschild solution, representing attractive candidates for realistic modeling of compact objects in nontrivial environments.

In this context, the interplay between external magnetic fields and surrounding dark matter is captured via an effective mass $\mu$ for an otherwise massless scalar field. Magnetic fields, which are prominent in galactic nuclei and accretion disks, supply a tunable environmental degree of freedom impacting both the energy spectrum and damping properties of QNMs. The scalar mass in the wave equation is therefore treated as an effective proxy for the field-environment interaction strength.

Mathematical Setup

The metric for the regular black hole is expressed as

$$ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega^2,$$

where $f(r) = 1 - 2m(r)/r$ and $m(r)$ is determined by the cumulative Einasto density:

$$\rho(r) = \rho_0 \exp\left[- \left(\frac{r}{h}\right)^{1/\tilde n}\right], \qquad m(r) = 4\pi \int_0^r x^2 \rho(x) dx.$$

For $\tilde n = 1/2$ and $1$, analytic expressions for the metric are available, facilitating high-fidelity spectral analysis. The background admits regularity at the origin, $$f(r) \to 1 - \frac{8\pi\rho_0}{3} r^2 + \mathcal{O}(r^3)$$, removing the curvature singularity and establishing a de Sitter–like core.

Linear scalar perturbations obey the modified Regge–Wheeler equation,

$$\frac{d^2 \Psi}{dr_*^2} + \left[\omega^2 - V(r)\right]\Psi = 0, \quad V(r) = f(r)\left( \frac{\ell(\ell+1)}{r^2} + \frac{f'(r)}{r} + \mu^2\right),$$

with QNM boundary conditions: ingoing at the horizon and outgoing at infinity.

Numerical Methodology

The QNM spectra are computed using high-order WKB approximations, up to 16th order for analytic backgrounds, supplemented by Padé resummation to improve convergence and accuracy. For regions where the WKB approach loses validity (e.g., near the breakdown of the potential barrier at large $\mu$0 and small $\mu$1), time-domain integration provides a robust cross-check for the dominant modes.

Grey-body factors (transmission coefficients) are calculated both via the WKB method and reconstructed from the lowest QNMs using established correspondences between the transmission probability and the QNM spectrum. The absorption cross-section is derived from partial transmission coefficients.

Spectral Properties: Trends and Regimes

A definitive trend observed is the monotonic suppression of the imaginary part of the QNM frequency $\mu$2 as the effective mass $\mu$3 increases, with a corresponding increase in $\mu$4. For example, with $\mu$5 and $\mu$6, $\mu$7 decreases by almost an order of magnitude as $\mu$8 is increased from 0 to 1, while $\mu$9 undergoes a substantial shift. This suppression of damping directly signals the approach to the quasi-resonance regime, where the black hole supports long-lived oscillatory modes. The critical value of $$ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega^2,$$0 at which the potential barrier disappears is $$ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega^2,$$1-dependent, and this structure is verified by extrapolating the computed spectra.

For moderate to large multipoles, deviations between different WKB orders and between WKB and time-domain results are quantitatively smaller than the changes induced by variations in $$ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega^2,$$2 or halo profile parameters, confirming the physicality of the observed spectral transitions.

Key numerical claims:

• For $$ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega^2,$$3, the extrapolated critical masses for the onset of extremely long-lived QNMs at $$ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega^2,$$4 are $$ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega^2,$$5, beyond which the effective potential loses its single-peak structure and the QNM boundary value problem qualitatively changes.
• The QNM spectra for regular black holes embedded in Einasto halos exhibit robust environmental sensitivity; modifications of $$ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega^2,$$6, $$ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega^2,$$7, or $$ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega^2,$$8 can induce spectral shifts exceeding those of all known regular black hole models surrounded by vacuum or phenomenological disks.

Grey-Body Factors and Scattering

Grey-body factors exhibit a low-frequency suppression that becomes more pronounced as $$ds^2 = -f(r)\,dt^2 + \frac{dr^2}{f(r)} + r^2 d\Omega^2,$$9 increases; the onset of efficient transmission is shifted to higher frequencies and the overall cross section transitions from the suppressed infrared domain to near-unity at high frequency in accord with geometric optics expectations. Both WKB and QNM-reconstruction techniques show close agreement, confirming the applicability of barrier-based methods for the examined regime.

By controlling the magnetic field, which directly modulates $f(r) = 1 - 2m(r)/r$0 via $f(r) = 1 - 2m(r)/r$1, the system's approach to the quasi-resonant regime can be tuned, and the observational signatures in ringdown and scattering can be mapped as a function of environmental parameters.

Astrophysical Implications and Future Directions

The results demonstrate that the combined effect of core regularity and physical environmental fields (magnetic and dark matter) allows for the realization of long-lived QNM ringdown signatures in realistic black hole models. This supports the program of probing both central regularity and environmental matter (e.g., dark matter profiles) via gravitational wave spectroscopy, building on recent EHT and LIGO/Virgo observations and projected PTA sensitivities.

Additionally, the parameter mapping between $f(r) = 1 - 2m(r)/r$2 and magnetic field strength provides a mechanism for constraining near-horizon electromagnetic conditions through QNM and grey-body measurements, offering new astrophysical diagnostics.

Future developments could include:

• Extension to rotating regular solutions matched to Einasto or other halo profiles;
• Direct linkage of observational ringdown data to dark matter profile parameters;
• Exploration of mode excitation and detectability with time-domain wave templates in realistic astrophysical scenarios;
• Generalization to higher spins, modified gravity, and nonlinear environmental interactions.

Conclusion

In summary, the study establishes the detailed quantitative connection between environmental matter fields, magnetic field-induced effective masses, and the QNM and scattering observables for regular black holes supported by Einasto halos. The emergence of long-lived, quasi-resonant black hole modes in these configurations is substantiated and shown to be tunable via cosmic field parameters, suggesting new observational probes for both core regularity and local environment in black hole astrophysics.

Reference: "Long-lived quasinormal frequencies for regular black hole supported by the Einasto profile in the presence of the magnetic field" (2603.28415).