Quantum instability of the Cauchy horizon in a charged de-Sitter spacetime with dark matter

The strong cosmic censorship conjecture (SCCC) requires that spacetime cannot be extended beyond the Cauchy horizon. This ensures the predictability of spacetime. In this paper, we investigate the SCCC for a spherically symmetric charged de-Sitter black hole surrounded by dark matter using classical and quantum scalar fields. At the classical level, we analyze the behavior of scalar waves near the Cauchy horizon using the method developed by Hintz and Vasy. We find a relationship between the Sobolev regularity of scalar waves and the spectral gap of quasinormal modes. In the nearly extremal region, this may lead to a violation of the SCCC. At the quantum level, we first provide a proof of the renormalizability of the quantum scalar field in dark-matter black holes. Using numerical methods, we then demonstrate that the renormalized quantum stress-energy tensor for any Hadamard state exhibits quadratic divergence near the Cauchy horizon in the nearly extremal region. The quadratic divergence of the renormalized quantum stress-energy tensor is sufficient to convert the Cauchy horizon into a singularity. Thus, the SCCC is preserved by quantum effects. Since the quadratic divergence is more singular than the behavior of classical scalar field perturbations near the Cauchy horizon, it means there is a region where physics is dominated by quantum effects. We study the influence of dark matter on quantum effects in this region and we find there is a monotonic relationship between the dark matter and the strength of quantum effects. The numerical results show that the quantum effects will become stronger as dark matter increases.