Improving Mimetic Gravity with Non-trivial Scalar Potential: Cosmology, Black Holes, Shadow and Photon Sphere

It is not easy to treat the spacetime with horizon(s) in the standard mimetic gravity. The solution to this problem has been presented in Phys.~Lett.~B 830 (2022), 137140, where it was suggested to modify the Lagrange multiplier constraint. In this paper, by using the improved formulation, we investigate the cosmology and black holes in mimetic gravity with scalar potential and in the scalar mimetic $F(R)$ gravity. The inflationary era and dark energy epoch for the above theories are presented as specific examples from the general reconstruction scheme which permits to realise any universe expansion history via the choice of the corresponding scalar potential or function $F(R)$. Two black hole solutions including the Schwarzschild and Hayward ones are constructed. The shadow and the radius of the photon sphere for the above black holes are found. The explicit confrontation of the black hole shadow radius with the observational bounds from M87$^*$ and Sgr A$^*$ objects is done. It is demonstrated that they do not conflict with Event Horizon Telescope observations.