Dynamical de Sitter black holes in a quasi-stationary expansion

We revisit and improve the analytic study arXiv:1804.03462 of spherically symmetric but dynamical black holes in Einstein's gravity coupled to a real scalar field. We introduce a series expansion in a small parameter $\epsilon$ that implements slow time dependence. At the leading order, the generic solution is a quasi-stationary Schwarzschild-de Sitter (SdS) metric, i.e. one where time-dependence enters only through the mass and cosmological constant parameters of SdS. The two coupled ODEs describing the leading order time dependence are solved up to quadrature for an arbitrary scalar potential. Higher order corrections can be consistently computed, as we show by explicitly solving the Einstein equations at the next to leading order as well. We comment on how the quasi-stationary expansion we introduce here is equivalent to the non-relativistic $1/c$ expansion.