Dynamical Tidal Response of Regular Black Holes: Perturbative Analysis and Shell EFT Interpretation

Abstract

We investigate the frequency-dependent (dynamical) tidal response of regular black holes for the Bardeen, Hayward, and Fan-Wang geometries. Our results are obtained by solving the coupled perturbation equations with appropriate boundary conditions, together with a `shell effective field theory' (EFT) construction in which the tidal response is encoded in renormalized, frequency-dependent response functions. In the polar sector, the frequency-dependent Love numbers exhibit strong dispersion, including oscillatory and resonant features, while smoothly recovering the static results in the zero-frequency limit. In the axial sector, where gravitational and electromagnetic perturbations remain coupled, the response shows a simpler but strongly frequency-dependent enhancement near extremality. The shell EFT analysis provides a gauge-invariant effective description of the tidal response and clarifies its renormalization structure, including the separation of scheme-independent logarithmic running and scheme-dependent finite contributions to the response coefficients, with the corresponding Wilson coefficients determined by matching to the black hole perturbation theory. Our results establish dynamical tidal Love numbers as well-defined EFT observables for regular black holes and show that they encode information about near-horizon and interior structure that is not accessible in the static limit.

Figures (9)

• Figure 1: Static TLN $\alpha(\ell_B)$ as a function of $\ell_B/\ell_{\rm ext}$ for the Bardeen black hole. The solid curve shows the numerical result obtained in the static limit, while the dashed curve corresponds to a best-fit polynomial $\alpha(\ell_B)=a\,\ell_B^2 + b\,\ell_B^4 + c\,\ell_B^6\,.$
• Figure 2: Dynamical Love number $\alpha(\omega)$ vs. frequency $\omega M$ for several fixed values of $\ell/\ell_{\rm ext}$ (Bardeen, polar sector).
• Figure 3: Dynamical Love number $\alpha(\omega)$ for the Bardeen geometry as a function of $\ell_B/\ell_{\rm ext}$ for several frequencies. The black dashed line shows the static Love number as obtained in Coviello:2025pla. For sufficiently small $\omega$ the response approaches the static curve and grows monotonically with $\ell_B/\ell_{\rm ext}$, while at larger $\omega$ the response can change sign at large $\ell_B/\ell_{\rm ext}$, illustrating the departure from the static response as dynamical effects become important.
• Figure 4: Dynamical Love number $\alpha(\omega)$ for the Hayward geometry as a function of $\ell_H/\ell_{\rm ext}$ for representative frequencies. The black dashed line shows the static Love number as obtained in Coviello:2025pla. For sufficiently small $\omega$ the response approaches the static curve and grows monotonically with $\ell_H/\ell_{\rm ext}$, while at larger $\omega$ the response departs from the static trend and can change sign at large $\ell_H/\ell_{\rm ext}$.
• Figure 5: Hayward frequency response $\alpha(\omega)$ as a function of $\omega M$ for several fixed values of $\ell_H/\ell_{\rm ext}$. The curves approach a constant in the low-frequency limit and develop oscillatory structure and localized extrema at intermediate frequencies, reflecting wave-interference and quasi-bound-state effects in the effective scattering problem.