Black Hole in the Expanding Universe with Arbitrary Power-Law Expansion

We present a time-dependent and spatially inhomogeneous solution that interpolates the extremal Reissner-Nordstr"om (RN) black hole and the Friedmann-Lema^itre-Robertson-Walker (FLRW) universe with arbitrary power-law expansion. It is an exact solution of the $D$-dimensional Einstein-“Maxwell”-dilaton system, where two Abelian gauge fields couple to the dilaton with different coupling constants, and the dilaton field has a Liouville-type exponential potential. It is shown that the system satisfies the weak energy condition. The solution involves two harmonic functions on a $(D-1)$-dimensional Ricci-flat base space. In the case where the harmonics have a single-point source on the Euclidean space, we find that the spacetime describes a spherically symmetric charged black hole in the FLRW universe, which is characterized by three parameters: the steepness parameter of the dilaton potential $n_T$, the U$(1)$ charge $Q$, and the “nonextremality” $\tau$. In contrast with the extremal RN solution, the spacetime admits a nondegenerate Killing horizon unless these parameters are finely tuned. The global spacetime structures are discussed in detail.