Test the weak cosmic censorship conjecture via black hole in dark matter halo

The weak cosmic censorship conjecture states that the black hole singularity is hidden inside the event horizon of the black hole, making it impossible for an external observer to measure. In this study, we investigate the weak cosmic censorship conjecture test of dark matter halo-black hole systems in both the cold dark matter model and ultralight dark matter model scenarios, with the aim of gaining insights into the influence of dark matter particles on the weak cosmic censorship conjecture. By examining the particle incident on an extremely or nearly extremal dark matter-black hole, as well as the scattering of a scalar field by an extreme or near-extreme dark matter-black hole. Our model calculations (based on second-order iterative approximation) indicate that under extremal conditions, the weak cosmic censorship conjecture may be violated in test particle scenarios; however, under near-extremal conditions, the relevant effects are all second-order small quantities, making it uncertain whether the conjecture is violated. Due to the influence of second-order small effects, the current approximation treatment cannot provide definitive conclusions. To obtain definitive conclusions, future studies need to incorporate self-gravitational effects, backreaction effects, and other important factors into more comprehensive theoretical models and conduct in-depth analysis, which is beyond the scope of this work. For the scalar field case, our model also suggests the possibility of the conjecture being violated under extremal conditions. In contrast, under near-extremal conditions, within the framework of our current approximation, the results are consistent with the view that the weak cosmic censorship conjecture holds. However, it must be emphasized that our research is based on second-order iterative approximation analysis, and whether the weak cosmic censorship conjecture would be violated if back-reaction effects and higher-order iterative corrections are fully considered still requires further in-depth research. This study contributes to deepening our understanding of the complex interaction mechanisms between dark matter and black holes.