Instability thresholds for de Sitter and Minkowski spacetimes in holographic semiclassical gravity

We study the stability of $d$-dimensional ($d=3,4,5$) de Sitter and Minkowski spacetimes within the framework of semiclassical gravity sourced by a strongly coupled quantum field with a gravity dual. Our stability results are derived from a careful analysis of the $d$-dimensional Lichnerowicz equation with mass-squared $m^2$ and of semiclassical equations involving the dimensionless parameter $\gamma_d$. For $d=3$, we find that Minkowski spacetime is always unstable against perturbations, whereas de Sitter spacetime becomes stable when a dimensionless parameter $\gamma_3$ exceeds a critical value. In $d=4$, both de Sitter and Minkowski spacetimes become unstable when the parameter $\gamma_4$ exceeds its critical value. In contrast, in $d=5$, de Sitter and Minkowski spacetimes remain stable for almost all values of the parameter $\gamma_5$, except for a regime in which higher-curvature corrections become comparable to the Einstein tensor.