Investigating strong gravitational lensing effects by suppermassive black holes with Horndeski gravity

We study gravitational lensing in strong-field limit by a static spherically symmetric black hole in quartic scalar field Horndeski gravity having additional hair parameter $q$, evading the no-hair theorem. We find an increase in the deflection angle $\alpha_D$, photon sphere radius $x_{ps}$, and angular position $\theta_{\infty}$ that increases more quickly while angular separation $s$ more slowly, but the ratio of the flux of the first image to all other images $r_{mag}$ decreases rapidly with increasing magnitude of the hair $q$. We also discuss the astrophysical consequences in the supermassive black holes at the centre of several galaxies and note that the black holes in Horndeski gravity can be quantitatively distinguished from the Schwarzschild black hole. Notably, we find that the deviation $\Delta\theta_{\infty}$ of black holes in Horndeski gravity from their general relativity (GR) counterpart, for supermassive black holes Sgr A* and M87, for $q=-1$ respectively, can reach as much as $25.192~\mu$as and $18.92~\mu$as while $\Delta s$ is about $1.121~\mu$as for Sgr A* and $0.8424~\mu$as for M87*. The ratio of the flux of the first image to all other images suggest that the Schwarzschild images are brighter than those of the black holes in Horndeski gravity, wherein the deviation $|\Delta r_{mag}|$ is as much as 3.082. The results suggest that observational tests of hairy black holes in Horndeski gravity are indeed feasible.