Weak and strong deflection gravitational lensings by a charged Horndeski black hole

Abstract: A charged black hole was predicted by the Einstein--Horndeski--Maxwell theory. In order to provide its observational signatures, we investigate its weak and strong deflection gravitational lensings. We find its weak deflection lensing observables, including the positions, magnifications and differential time delay of the lensed images. We also obtain its strong deflection lensing observables, including the apparent radius of the photon sphere as well as the angular separation, brightness difference and differential time delay between the relativistic images. Taking the supermassive black hole in the Galactic Center as the lens, we evaluate these observables and compare these signatures with those of the Schwarzschild, Reissner-Nordstr\"{o}m, tidal Reissner-Nordstr\"{o}m and charged Galileon black holes. After a detailed analysis of the feasibility of measuring these lensing observables, we conclude that although it is possible to detect some leading effects of the weak and strong deflection lensings by the charged Horndeski and other black holes with current technology, it would be unlikely to distinguish one kind of these black holes from the others based on these detections in the near future due to lack of enough highly angular resolution in astronomical observations to tell their differences.