Non-Minimally Coupled Electromagnetic Fields and Observable Implications for Primordial Black Holes

General relativity (GR) postulates have been verified with high precision, yet our understanding of how gravity interacts with matter fields remains incomplete. Various modifications to GR have been proposed in both classical and quantum realms to address these interactions within the strong gravity regime. One such approach is non-minimal coupling (NMC), where the space-time curvature (scalar and tensor) interacts with matter fields, resulting in matter fields not following the geodesics. To probe the astrophysical implications of NMC, in this work, we investigate non-minimally coupled electromagnetic (EM) fields in the presence of black holes. Specifically, we show that primordial black holes (PBHs) provide a possible tool to constrain the NMC parameter. PBHs represent an intriguing cosmological black hole class that does not conform to the no-hair theorem. We model the PBH as a Sultana–Dyer black hole and compare it with Schwarzschild. We examine observables such as the radius of the photon sphere, critical impact parameter, and total deflection angles for non-minimally coupled photons for Schwarzschild and Sultana–Dyer black holes. Both the black hole space-times lead to similar constraints on the NMC parameter. For a PBH of mass M=10−5M⊙, the photon sphere will not be formed for one mode. Hence, the photons forming the photon sphere will be highly polarized, potentially leading to observable implications.