Black holes, photon sphere, and cosmology in ghost-free $f(\mathcal{G})$ gravity

The improved version of ghost-free $f(\mathcal{G})$ gravity introduced in Phys.~Lett.~B \textbf{631} (2005), 1-6 is proposed and investigated. It is demonstrated that improved ghost-free $f(\mathcal{G})$ gravity may be consistently applied to describe the expanding universe (with horizon) as well as the static spacetime like black holes. Interestingly, such a theory looks like a close avatar of scalar-Einstein-Gauss-Bonnet gravity with the only difference that the scalar field is not a dynamical one so that many predictions of ghost-free $f(\mathcal{G})$ gravity are very similar to that of sEGB gravity. We discuss the gravitational wave in the improved model with the condition that the propagating speed of the gravitational wave is identical to that of the propagation speed of light. A model that describes inflation in the early universe and could satisfy the observational constraints is also constructed. The solution of ghost-free $f(\mathcal{G})$ gravity realising the given static spacetime with spherical symmetry is proposed. Some of such solutions realise explicitly the Reissner-Nordstr\"{o}m black hole or the Hayward black hole, which is a regular black hole without curvature singularity, We also construct a black hole in our theory which contains the Arnowitt-Deser-Misner (ADM) mass, the horizon radius, and the radius of the photon sphere as independent parameters. The radius of the black hole shadow in this model as well as photon sphere radius are estimated. It is shown that there exists a parameter region which satisfies the constraints coming from Event Horizon Telescope observations. Furthermore, we construct model where the radius of the black hole shadow or photon orbit is smaller than the horizon radius supposed from the ADM mass.