Gravitational lensing and shadows of dilatonic black holes in dilaton-massive gravity

This paper presents optical and astrophysical aspects of charged dilatonic black holes within the framework of dilaton-massive gravity, a promising extension of General Relativity motivated by low-energy string theory. By incorporating a nonminimally coupled scalar dilaton field and massive graviton terms into the Einstein-Maxwell action, we study static, spherically symmetric black hole solutions characterized by the parameters c 1 , c 2 , and m 0 , which control the strength of scalar-gravity and massive gravity couplings. We use Gauss-Bonnet theorem to compute the weak deflection angle of light around the black hole. The results reveal that the coupling parameter c 1 intensifies light bending, while more negative values of c 2 and higher graviton masses m 0 suppress it. Extending this analysis to realistic astrophysical settings, we introduce a cold, non-magnetized plasma environment and derive the plasma-corrected deflection angle. The presence of plasma leads to chromatic dispersion, increasing the bending of light at lower frequencies, and amplifying the influence of dilaton-massive gravity on lensing observables. Next, we examine the black hole's shadow by numerically solving for the photon sphere and evaluating the shadow radius. Our findings demonstrate that larger values of c 1 and m 0 decrease the shadow size due to increased spacetime curvature, whereas increasing c 2 expands it by lowering the gravitational potential. These effects are visualized through shadow images in celestial coordinates, clearly illustrating how modified gravity alters the observable structure of black holes.