Exploring black holes with multiple photon spheres by interferometric signatures

In this paper, we investigate the interferometric signatures of hairy Schwarzschild black holes (hSBHs) that have either single or double photon spheres. Our interest mainly stems from two considerations: (i) the photon ring structure in black hole images produces strong and universal interferometric signatures on long baselines, enabling precision measurements of black hole parameters and testing gravitational theory; (ii) the hSBH describes the deformation of standard Schwarzschild black hole induced by additional sources, and they can feature double photon spheres within certain parameter regimes. Using both analytical and numerical methods, we find that for a hSBH with a single photon sphere the complex visibility amplitude of the image exhibits damped oscillations. Similar behavior appears in the double photon sphere case when the inner photon sphere has lower effective potential than the outer one, as the photons near the inner photon sphere remain trapped by gravity. However, when the inner potential is higher, a beat pattern rises. Our findings reveal that the complex visibility amplitude can encode the signature of the photon sphere structure of the central black hole.