Recent times witnessed a surge of interest in strong gravitational lensing by black holes due to the Event Horizon Telescope (EHT) results, which suggest comparing the black hole lensing in general relativity and modified gravity theories. This 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 from surrounding fluid, like dark matter, having conserved energy-momentum tensor. These hairy Kerr black holes may also 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 a deviation parameter $\alpha$ and a primary hair ${\ell }_0$. Interestingly, the deflection coefficient $\overline{a}$ increases and decreases with increasing ${\ell }_0$ and $\alpha$, respectively. $\overline{b}$ shows opposite behavior with ${\ell }_0$ and $\alpha$. We also find that the deflection angle ${\alpha }_D$, angular position ${\theta }_{\infty }$, and $u_m$ decrease, but angular separation $s$ increases with $\alpha$. We compare our results with those for Kerr black holes, and also apply the formalism 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 $2.6\mu \mathrm{as}$ for Sgr A* and $1.96\mu \mathrm{as}$ for M87*, which are unlikely to be resolved by the current EHT observations.

中文翻译：

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毛克尔黑洞的强场引力透镜

由于事件视界望远镜（EHT）的结果，最近见证了对黑洞强引力透镜的兴趣激增，这表明将广义相对论和修正引力理论中的黑洞透镜进行比较。这可能有助于我们评估这些模型之间的现象学差异。克尔黑洞也是一些替代引力理论的解决方案，而最近获得的修正克尔黑洞（毛克尔黑洞）避开了无毛定理，是由于来自周围流体的额外来源，如暗物质，具有守恒的能量-动量张量。这些毛茸茸的克尔黑洞也可能是另一种引力理论的解决方案。我们将先前在强偏转限制中的克尔黑洞引力透镜的工作推广到毛茸茸的克尔黑洞，$\alpha$ 和一个初级头发 ${\ell }_0$. 有趣的是，挠度系数$\overline{\mathrm{一种}}$ 随着增加而增加和减少 ${\ell }_0$ 和 $\alpha$， 分别。 $\overline{乙}$ 表现出相反的行为 ${\ell }_0$ 和 $\alpha$. 我们还发现偏转角${\alpha }_D$, 角位置 ${\theta }_{\infty }$， 和 ${你}_{米}$ 减少，但角度分离 $秒$ 增加 $\alpha$. 我们将我们的结果与克尔黑洞的结果进行比较，并应用形式主义来讨论超大质量黑洞 Sgr A* 和 M87* 背景下的天体物理学后果。我们观察到角位置与克尔黑洞角位置的偏差不超过$2.6\mu \mathrm{作为}$ 对于 Sgr A* 和 $1.96\mu \mathrm{作为}$ 对于 M87*，目前的 EHT 观测不太可能解决。