Black holes in the Universe: Generalized Lemaître-Tolman-Bondi solutions

We present new exact solutions which presumably describe black holes in the background of a spatially flat, pressureless dark-matter– or dark matter plus dark energy ($\mathrm{DM}+\mathrm{DE}$)- or quintom-dominated Universe. These solutions generalize Lemaître-Tolman-Bondi metrics. For a dark-matter– or ($\mathrm{DM}+\mathrm{DE}$)-dominated universe, the area of the black hole apparent horizon (AH) decreases with the expansion of the Universe while that of the cosmic AH increases. However, for a quintom-dominated universe, the black hole AH first shrinks and then expands, while the cosmic AH first expands and then shrinks. A ($\mathrm{DM}+\mathrm{DE}$)-dominated universe containing a black hole will evolve to the Schwarzschild-de Sitter solution with both AHs approaching constant size. In a quintom-dominated universe, the black hole and cosmic AHs will coincide at a certain time, after which the singularity becomes naked, violating cosmic censorship.

Figure 1

The size of the black hole AH (lower curve) in a dust-dominated universe decreases with the cosmic expansion, while that of the cosmic AH (upper curve) increases. The MS mass of the black hole tends to zero in the future. There was a time at which the two horizons coincided and before which they were absent and the singularity was naked. The plots correspond to the parameter values $m=1$, ${\rho }_0=0.05$ in Planck units. In the International Units system, they are $m={10}^{24}{\mathrm{M}}_{\odot }$ and ${\rho }_0=0.05·{10}^{-123}{\rho }_P$ (${\rho }_P$ is the Planck density).

Figure 2

The AH of an astronomical black hole in a dust-dominated universe decreases with the cosmic expansion. The plots correspond to the parameter value $m={10}^7{\mathrm{M}}_{\odot }$. The units of $x$ and $t$ are $2m$ (the corresponding Schwarzschild radius) and ${\mathrm{H}}_0^{-1}$, respectively.

Figure 3

In a $\Lambda \mathrm{CDM}$ universe, the black hole AH (lower curve) shrinks while the cosmic AH (upper curve) expands with the cosmic expansion. There was a time at which the two horizons coincided and before which no horizon existed and the singularity was naked. Late in the history of the Universe, both black hole and cosmic AHs approach constant size. The plots correspond to the parameter values $m=2.2·{10}^{22}{\mathrm{M}}_{\underset{}{\overset{}{⨀}}}$, ${\rho }_0=0.27{\rho }_C$, ${\rho }_{\Lambda }=0.73{\rho }_C$. ${\rho }_C$ is the current cosmic density. The unit of $t$ is the present Hubble time $H_0^{-1}$.

Figure 4

In a quintom-dominated universe, the black hole AH (lower curve) first shrinks and then expands, while the cosmic AH (upper curve) first expands and then shrinks. There exist two instants of time, one in the past and one in the future, at which the two horizons coincide. Before or after these critical instants, no AH exists and a naked singularity is present. The plots correspond to the parameter values $m=1$, ${\rho }_0=0.0002$, $p_0=0.001$, and $t_0=1$.