Chaotic solutions and black hole shadow in f(R) gravity

We discuss the emergence of black hole shadow and photon-sphere in the context of f(R) gravity. It is shown that the shadow is exponentially sensitive to linear instabilities of metric coming from some f(R) solutions. Thus, the instabilities of photon circular trajectories, delimiting the black hole photon-sphere, are double exponentialized. Specifically we individuate two Lyapunov exponents, rather than only one, related to two different sources of chaos in geodesic orbits as a sort of butterfly effect. Such a result violates the black hole chaos bound proposed by Maldacena, Shenker and Stanford for General Relativity. We also explore the impact of the black hole metric instabilities in f(R) gravity on the quasi-normal modes. In the framework of Extended Theories of Gravity, our analysis suggests a new paradigm to deal with black hole shadow and gravitational waves observations coming from black hole merging in the ringdown phase.The work of A.A. is supported by the Talent Scientific Research Program of College of Physics, Sichuan University, Grant No. 1082204112427. S.C. acknowledge the support of Istituto Nazionale di Fisica Nucleare (INFN) (iniziative specifiche MOONLIGHT2 and QGSKY)