Strong cosmic censorship in de Sitter spacetimes with dark matter

After imposing the weak energy condition on dark matter, we find that even in the absence of charge, a spherically symmetric de Sitter black hole with perfect fluid dark matter may still possess a Cauchy horizon. This paper investigates the stability of the Cauchy horizon in such a black hole under perturbations from a massless scalar field. Through numerical calculations of the massless scalar field's quasinormal modes, we discover that in this spacetime, when the black hole approaches extremality, i.e., when the dark-matter parameter b approaches its maximum value $b_{\text{max}}$, the Strong Cosmic Censorship (SCC) conjecture may be violated. This is the first instance of classical SCC violation observed in a spherically symmetric, uncharged black hole. Additionally, we explore the region of SCC violation within the parameter space $\Lambda M^2-b/b_{\text{max}}$. Our findings indicate that as the cosmological constant increases, the violation region initially expands and then contracts.