Super-horizon primordial black holes

We discuss a new class of solutions to the Einstein equations which describe a primordial black hole (PBH) in a flat Friedmann background. Such solutions arise if a Schwarzschild black hole is patched onto a Friedmann background via a transition region. They are possible providing the black hole event horizon is larger than the cosmological apparent horizon. Such solutions have a number of strange features. In particular, one has to define the black hole and cosmological horizons carefully and one then finds that the mass contained within the black hole event horizon decreases when the black hole is larger than the Friedmann cosmological apparent horizon, although its area always increases. These solutions involve two distinct future null infinities and are interpreted as the conversion of a white hole into a black hole. Although such solutions may not form from gravitational collapse in the same way as standard PBHs, there is nothing unphysical about them, since all energy and causality conditions are satisfied. Their conformal diagram is a natural amalgamation of the Kruskal diagram for the extended Schwarzschild solution and the conformal diagram for a black hole in a flat Friedmann background. In this paper, such solutions are obtained numerically for a spherically symmetric universe containing a massless scalar field, but it is likely that they exist for more general matter fields and less symmetric systems.

Figure 1

The conformal diagram for a flat Friedmann spacetime with a massless scalar field. The cosmological apparent horizon, which is a past outer trapping horizon (POTH), is spacelike and outside the cosmological particle horizon (CPH).

Figure 2

The conformal diagram for a PBH smaller than the Friedmann cosmological particle horizon.

Figure 3

Schematic figures for the initial data corresponding to PBHs (a) smaller and (b) larger than the cosmological apparent horizon (CAH).

Figure 4

Positions in ($u$, $v$) plane of black hole event horizon (BHEH) and trapping horizons with $r_{,v}=0$ and $r_{,u}=0$, plotted with solid, dashed, and dotted-dashed lines, respectively. The future and past outer trapping horizons are labeled as “FOTH” and “POTH,” respectively. The signs of $(r_{,v})$ and $(r_{,u})$ are also shown. A region is future trapped if it has ($-$, $-$), past trapped if it has ($+$, $+$), and untrapped if it has ($+$, $-$) or ($-$, $+$).

Figure 5

The area radii of the black hole event horizon (BHEH) and trapping horizons with $r_{,v}=0$ and $r_{,u}=0$, plotted with solid, dotted, and dashed lines, respectively. We can see the area radius of the black hole event horizon always increases.

Figure 6

The Misner-Sharp masses for the black hole event horizon (BHEH) and trapping horizons with $r_{,v}=0$ and $r_{,u}=0$, plotted with solid, dotted, and dashed lines, respectively. The mass of the black hole event horizon decreases when it is in a past trapped region and increases when it is in an untrapped region with null expansions ($+$, $-$). The mass of the trapping horizon with $r_{,v}=0$ is slightly larger than that of the black hole event horizon for each model, although they are almost indistinguishable.

Figure 7

Dependence of mass increase rate $d{m}_{\mathrm{BHEH}}/dv$ of the black hole event horizon on $v$ for Models E–J. It is negative when the black hole event horizon is in the past trapped region but positive when it gets out of the past trapped region.

Figure 8

The evolution of $r$ along (a) $v=\mathrm{const}$ and (b) $u=\mathrm{const}$ and of $2m/r$ along (c) $v=\mathrm{const}$ and (d) $u=\mathrm{const}$ for Model F. In (a) and (c), the arrow at $u=-0.5$ denotes the matching surface, while the arrow at $u\simeq 0.400$ denotes the black hole event horizon.

Figure 9

Conformal diagram showing the possible causal structure for a PBH larger than the Friedmann cosmological apparent horizon. There are two distinct future null infinities and a past trapped (white hole) region is converted into a future trapped (black hole) region. The region with $u<u_{\mathrm{m}}$ is the usual flat Friedmann spacetime but the region $v<v_0$ is not calculated. See text for details.

Figure 10

Conformal diagram of the maximally extended Schwarzschild spacetime, i.e., the Kruskal diagram. The black hole event horizon (BHEH) coincides with the future outer trapping horizon and the white hole event horizon (WHEH) coincides with the past outer trapping horizon.

Figure 11

Conformal diagram of a possible causal structure of the maximally extended spacetime for a PBH larger than the Friedmann cosmological apparent horizon. There are two distinct future null infinities, corresponding to the conversion of a past trapped (white hole) region into a future trapped (black hole) region.

Figure 12

Conformal diagram of the critical PBH spacetime, where the black hole event horizon (BHEH) coincides with the cosmological particle horizon (CPH).