Connection Between the Shadow Radius and Quasinormal Frequencies for Black Holes in STVG with Perfect Fluid Dark Matter

TL;DR

This paper establishes a precise analytical relation between black hole shadow radius and quasinormal mode frequencies in scalar-tensor-vector gravity with dark matter, validated by numerical methods, linking gravitational ringdown and shadow observations.

Contribution

It derives an exact analytical connection between shadow radius and QNM frequencies in STVG with dark matter, validated numerically for various perturbations and multipole numbers.

Findings

The real part of QNM frequencies increases with dark matter intensity and decreases with the MOG parameter.

An exact relation between shadow radius and QNM frequency is derived in the eikonal limit.

Numerical results confirm the analytical relation at large multipole numbers.

Abstract

We investigate the connection between black hole shadow and quasinormal mode (QNM) spectra in the context of scalar-tensor-vector gravity (STVG) coupled to perfect fluid dark matter (PFDM), characterized by the MOG parameter $\alpha$ and the dark matter intensity $\lambda$. Employing complementary methods -- namely the sixth-order WKB approximation, Pad\'e resummation, and time-domain numerical integration -- we compute QNM frequencies for scalar ($s=0$), electromagnetic ($s=1$), and axial gravitational ($s=2$) perturbations. Both the real part of the QNM frequencies and the peak height of the effective potential display a consistent parametric dependence: they increase with $\lambda$ yet decrease with growing $\alpha$. In the eikonal limit ($l \gg 1$), we derive an exact analytical link between the shadow radius $R_{\mathrm{sh}}$ and the QNM frequency $\omega_R$. Noting that $R_{\mathrm{sh}}$ is determined by the critical impact parameter $b_c = r_{\mathrm{ph}}/\sqrt{f(r_{\mathrm{ph}})}$, while $\omega_R = \Omega l$ with photon angular velocity $\Omega = \sqrt{f(r_{\mathrm{ph}})}/r_{\mathrm{ph}}$, we obtain the precise relation $\omega_R = l / b_c$, identifying $R_{\mathrm{sh}} \equiv b_c$ for an asymptotically flat observer. This prediction is robustly validated by numerical results across all three computational approaches at large multipole numbers. Our findings reveal that the black hole shadow and gravitational ringdown are not independent phenomena, but dual observational signatures of the same underlying structure -- the unstable photon orbit -- thereby offering a unified multi-messenger framework to simultaneously constrain modified gravity and dark matter in the strong-field regime.