Strong field gravitational lensing by hairy Kerr black holes

The recent time witnessed a surge of interest in strong gravitational lensing by black holes due to the Event Horizon Telescope (EHT) results, suggesting comparing the black hole lensing in general relativity and modified gravity theories. That may help us to assess the phenomenological differences between these models. A Kerr black hole is also a solution to some alternative theories of gravity, while recently obtained modified Kerr black holes (hairy Kerr black holes), which evade the no-hair theorem, are due to additional sources the surrounding fluid, like dark matter, having conserved energy momentum tensor (EMT). These hairy Kerr black holes may be solutions to an alternative theory of gravity. We generalize previous work on gravitational lensing by a Kerr black hole, in the strong deflection limits to the hairy Kerr black holes, with deviation parameter $\alpha$ and a primary hair $\ell_0$. Interestingly, the deflection coefficient $\bar{a}$, respectively, increases and decreases with increasing $\ell_0$ and $\alpha$. $\bar{b}$ shows opposite behaviour with $\ell_0$ and $\alpha$. We also find that the deflection angle $\alpha_D$, angular position $\theta_{\infty}$ and $u_{m}$ decreases, but angular separation $s$ increases with $\alpha$. We compare our results with those for Kerr black holes, and also, the formalism is applied to discuss the astrophysical consequences in the context of the supermassive black holes Sgr A* and M87*. We observe that the deviations of the angular positions from that of the Kerr black hole are not more than $2.6~\mu$as for Sgr A* and $1.96~\mu$as for M87*, which are unlikely to get resolved by the current EHT observations.