Black Holes in an Expanding Universe

An exact solution representing black holes in an expanding universe is found. The black holes are maximally charged and the universe is expanding with arbitrary equation of state ($P=w\rho$ with $-1\le \forall w\le 1$). It is an exact solution of the Einstein-scalar-Maxwell system, in which we have two Maxwell-type U(1) fields coupled to the scalar field. The potential of the scalar field is an exponential. We find a regular horizon, which depends on one parameter [the ratio of the energy density of U(1) fields to that of the scalar field]. The horizon is static because of the balance on the horizon between gravitational attractive force and U(1) repulsive force acting on the scalar field. We also calculate the black hole temperature.

Figure 1

Two horizon radii (${\tilde{R}}_{-}$ and ${\tilde{R}}_{+}$ or ${\tilde{R}}_H$) for $n_T=1$ and 3. ${\tilde{R}}_H$ and ${\tilde{R}}_C$, which is not a cosmological horizon, degenerate at ${\tau }_{\mathrm{cr}}=3\sqrt{3}/2$.

Figure 2

Black hole temperatures on two horizons (${\tilde{R}}_{-}$ and ${\tilde{R}}_{+}$ or ${\tilde{R}}_H$) for $n_T=1$ and 3. ${\tilde{T}}_{\mathrm{BH}}=Q{T}_{\mathrm{BH}}$.