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

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, Padé 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 $R_{\mathrm{s}\mathrm{h}}$ and the QNM frequency ${\omega }_R$. Noting that $R_{\mathrm{s}\mathrm{h}}$ is determined by the critical impact parameter $b_c=r_{\mathrm{p}\mathrm{h}}\mathrm{/}\sqrt{f(r_{\mathrm{p}\mathrm{h}})}$, while ${\omega }_R=\mathrm{\Omega }l$ with photon angular velocity $\mathrm{\Omega }=\sqrt{f(r_{\mathrm{p}\mathrm{h}})}\mathrm{/}{r}_{\mathrm{p}\mathrm{h}}$, we obtain the precise relation ${\omega }_R=l\mathrm{/}{b}_c$, identifying $R_{\mathrm{s}\mathrm{h}}\equiv b_c$ 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.