Influences of tilted thin accretion disks on the observational appearance of hairy black holes in Horndeski gravity

Research on the observational appearance of black holes, both in general relativity and modified gravity, has been in full swing since the Event Horizon Telescope Collaboration announced photos of M87$^{*}$ and Sagittarius A$^{*}$. Nevertheless, limited attention has been given to the impact of tilted accretion disks on black hole images. This paper investigates the $230$ GHz images of non-rotating hairy black holes illuminated by tilted, thin accretion disks in Horndeski gravity with the aid of a ray tracing method. The results indicate that reducing the scalar hair parameter effectively diminishes image luminosity and extends both the critical curve and the inner shadow. This trend facilitates the differentiation between hairy black holes and Schwarzschild black holes, especially in certain parameter spaces where the current Event Horizon Telescope array is capable of capturing such variations. Furthermore, we observe that the inclination of the tilted accretion disk can mimic the observation angle, consequently affecting image brightness and the morphology of the inner shadow. In specific parameter spaces, alterations in the tilt or position of the accretion disk can lead to a drift in the light spot within the images of hairy black holes. This finding may establish a potential correlation between the precession of the tilted accretion disk and image features. Additionally, through an examination of images depicting hairy black holes surrounded by two thin accretion disks, we report the obscuring effect of the accretion environment on the inner shadow of the black hole.