Weak gravitational lensing of black holes in generalized cosmological models with k-essence

This article considers the extension of the general theory of relativity by introducing additional geometric invariants in the action, which leads to a modified black hole metric. Based on the model with a metric-affine approach, an analytical analysis of gravitational lensing and the formation of a black hole shadow in a plasma medium is performed. The influence of weak gravitational lensing of black holes is investigated within the framework of cosmological models extended by scalar fields of -essence. Models with -essence, nonlinear kinetic dynamics, are attractive candidates for explaining the accelerated expansion of the universe without introducing a cosmological constant. Detailed formulas are provided that describe the structure of the horizon, the photon sphere, the angle of light deviation, and the lensing equations. Within such models, the deformation of the image of light sources passing near black holes was analyzed, taking into account the change in the effective metric. The modifications of the deflection angle and the distribution of shifts arising due to weak lensing were considered. The results obtained show that -essence fields can lead to observable corrections to the standard types of general relativity, which opens up new prospects for using gravitational lensing as a tool for testing the fundamental properties of dark energy and modified gravity in astrophysical observations.