Probing Horndeski Gravity via Kerr Black Hole: Insights from Thin Accretion Disks and Shadows with EHT Observations

In this study, we have considered the Kerr-like black hole (BH) model in Horndeski gravity and analyse the visual characteristics of shadow images under two illumination models, such as a celestial light source and a thin accretion disk. To capture the BH shadow images, we utilise a recent fisheye camera model and ray-tracing procedures. In this view, we carefully addressed the influence of the spin parameter $a$ and the hair parameter $h$ on the BH shadow images. The results indicate that for smaller values of $h$, the BH shadow contours shift noticeably towards the right side of the screen, while for larger values of $h$, the nearly circular shadow gradually deforms into a ``D'' shape profile. For a celestial light source, the larger values of $h$ lead to a reduction in the corresponding radius of the photon ring, while the space-dragging effect becomes more pronounced with increasing $a$. We further discuss the distinctive characteristics of images observed in both prograde and retrograde accretion disk scenarios. The results reveal that variations in $h$ significantly affect both the inner shadow and the resulting Einstein ring. Subsequently, we also discussed the distinct features of red-shift configurations on the disk for both direct and lensed images, which are closely related to the accretion flow and the relevant parameters. We also attempt to use the recent observational data from M$87^{\ast}$ and Sgr $A^{\ast}$ and constrain the hair parameter $h$, confirming the validity of Horndeski gravity.