Universality of Quasinormal-Mode Shifts from Small Nonlocal Effective Couplings

TL;DR

This work develops a perturbative framework for computing first-order quasinormal-mode shifts caused by small nonlocal fractional perturbations of the wave operator, modeled by $\delta\mathcal{L}=\ell_{\rm nl}^{2s}(-\Delta)^s$ with $0<s<1$. The central result is a universal scaling law for the fractional QNM shift, $\frac{\delta\omega}{\omega_0} = -\frac{\varepsilon}{2\omega_0^2}\frac{\langle \tilde{\psi}_0|\delta\mathcal{L}|\psi_0\rangle}{\langle \tilde{\psi}_0|\psi_0\rangle}$, where $\delta\mathcal{L}$ encodes nonlocality and the geometry enters only through overlap integrals of the unperturbed and dual modes. Across Schwarzschild, slowly rotating Kerr, Hayward, and LQG-corrected black holes, the leading-order shifts share the same functional form, with an additional universal $\ell^{2s}$ enhancement in the eikonal limit $\ell\gg1$. This universality provides a model-independent signature of nonlocality in strong-gravity ringdown spectra and offers a potential observational window into quantum-gravity-inspired modifications, especially as gravitational-wave measurements of BH ringdowns improve. The framework connects nonlocal EFTs and UV-complete gravity to observable QNM spectra and can be extended to broader perturbations and higher-order operators.

Abstract

We investigate perturbative quasinormal-mode (QNM) shifts of black holes arising from fractional, nonlocal modifications to the wave operator. Starting from a scalar master equation corrected by a small fractional Laplacian term $(-Δ)^{s}$ with $0<s<1$, we derive an analytic expression for the complex frequency shift at first order in the nonlocal coupling $\varepsilon$. Evaluation of the fractional operator in both coordinate and momentum representations reveals a universal scaling law $δω/ω\propto \varepsilon/M^{2s}$, largely independent of the field spin, with an additional $\ell^{2s}$ enhancement in the eikonal regime $\ell \gg 1$. Applying the formalism to Schwarzschild, slowly rotating Kerr, Hayward regular, and LQG-corrected black holes, we demonstrate that the leading-order fractional QNM shift is universal, with geometric details entering only through overlap integrals of the mode functions. This universality provides a model-independent signature of nonlocality in strong-gravity ringdown spectra and offers a potential observational window into quantum-gravity-inspired modifications.