Separate universes do not constrain primordial black hole formation

Carr and Hawking showed that the proper size of a spherical overdense region surrounded by a flat Friedmann Robertson Walker (FRW) universe cannot be arbitrarily large as otherwise the region would close up on itself and become a separate universe. From this result, they derived a condition connecting size and density of the overdense region ensuring that it is part of our universe. Carr used this condition to obtain an upper bound for the density fluctuation amplitude with the property that for smaller amplitudes the formation of a primordial black hole is possible, while larger ones indicate a separate universe. In contrast, we find that the appearance of a maximum is not a consequence of avoiding separate universes but arises naturally from the geometry of the chosen slicing. Using instead of density a volume fluctuation variable reveals that a fluctuation is a separate universe if this variable diverges on superhorizon scales. Hence, Carr’s and Hawking’s condition does not pose a physical constraint on density fluctuations. The dynamics of primordial black hole formation with an initial curvature fluctuation amplitude larger than the one corresponding to the maximum density fluctuation amplitude was previously not considered in detail and so we compare it to the well-known case where the amplitude is smaller by presenting embedding and conformal diagrams of both types in dust spacetimes.

Figure 1

Sketch of an idealized fluctuation: flat FRW for $r>r_b$ and closed FRW for $r<r_a$.

Figure 2

Comparison of $r$ and $\chi$ radial coordinates. Both circles show an embedded 3-sphere ($\theta$ and $\varphi$ directions suppressed).

Figure 3

Embeddings of spatial slices of different types of density fluctuations (angular dimensions suppressed). The left figure shows a case where $\zeta <{\zeta }_{\max }$ (${\chi }_a<\pi /2$, $\delta <{\delta }_{\max }$) called type I, the middle figure shows the case $\zeta ={\zeta }_{\max }$ (${\chi }_a=\pi /2$, $\delta ={\delta }_{\max }$) and the right one shows a case where $\zeta >{\zeta }_{\max }$ (${\chi }_a>\pi /2$, $\delta <{\delta }_{\max }$) called type II.

Figure 4

Comparison of $\overline{\zeta }$, $\zeta$ and $\delta$.

Figure 5

Mass and energy functions. Full lines correspond to type I and dashed lines to type II.

Figure 6

$t=0.1$ Since the Hubble horizon is tiny (there is only a small black region in the center) the spatial shapes are nearly unchanged at this early time compared to its initial values at $t_i=0$. Both extrinsic curvatures $K^{\widehat{\chi }}$ and $H_R$ are very large (long arrows) and positive (outward-directed arrows) corresponding to the small Hubble radius. The arrows are actually 30 times longer and were rescaled to fit the paper. The following pictures however show the correct length. top: Two outgoing photons, the yellow and orange dots (light grey and grey for a b/w print) start in the middle. bottom: One outgoing orange photon starts in the middle.

Figure 7

$t=1.1$ top: A third blue photon traveling inward appeared. But it is trapped and hence recedes as long as the embedding at its position is orange. bottom: The Hubble horizon overtakes the orange outgoing photon.

Figure 8

$t=2.1$ top: Horizon crossing occurred and the neck starts to stretch: the proper distance of the empty region between the black dots grows. The extrinsic curvature $K$ gets smaller, but in the neck it remains larger since it has to create the negative intrinsic curvature ${}^{(3)}R$. bottom: A third green photon traveling inwards appeared.

Figure 9

$t=3.1$ top: The outer-trapped orange region gets fragmented: vertical regions naturally remain trapped for a longer time: Moving in these regions does not alter $R$ much. bottom: The orange outgoing photon now overtakes the Hubble horizon.

Figure 10

$t=4.1$ top: Note that type II fluctuations with their local minimal $R$ may be seen as wormholes according to the definition C of 65. To prevent a static wormhole from becoming a black hole one needs exotic matter violating a few energy conditions (see 62). In an accelerating universe, a dynamical superhorizon-sized wormhole will never reenter the horizon and hence does not require exotic matter to be stable. In our decelerating dust Universe, we would also need exotic matter to stabilize the throat once it entered the horizon. Since we do not have exotic matter, we see that just below the orange photon at the neck, a tiny red future trapped region appeared. This means that the mass of the wormhole got larger than half its Schwarzschild radius. Once this happened, the wormhole ceases to be traversable (in the sense that one cannot return) and everything below the outer boundary between red and black—the black hole apparent horizon—is doomed. $K^{\widehat{\chi }}$ in the neck is still positive (green) but $H_R$ became negative in the neck center which turned the outer-trapped region into an inner-trapped one (see (44)). Note also that the formation of the trapped region occurred bottom: The orange photon leaves the scene. No BH horizon has appeared yet.

Figure 11

$t=5.1$ top: The orange photon nearly rests above BH horizon. The trapped contracting red region grows due to the stretching of the wormhole driven by the positive $K^{\widehat{\chi }}$ (green). bottom: The blue photon will soon reach the center.

Figure 12

$t=6.1$ The visible region is now completely within the Hubble horizon. top: Also the outer-trapped region in the bubble is disappearing. The blue and yellow photon bravely march towards inevitable oblivion.

Figure 13

$t=7.1$ top: At maximum expansion $t_m$ the sign of $H_R$ changed in the bubble (orange $\to$ red), where a new future trapped horizon has appeared. This is the essential horizon for a type I fluctuation. In the neck, only $H_R$ changed sign and $K^{\widehat{\chi }}$ remained positive (green) such that the second term in (42) becomes positive and (visibly) reduces the negative ${}^{(3)}R$ in the neck (see (41)). This also leads to a direction-dependent contraction/expansion of space. Since in the lower hemisphere $K^{\widehat{\chi }}$ got negative, space contracts there in every direction. bottom: Here, the sign of $H_R$ and $K^{\widehat{\chi }}$ changed in the whole closed part (which is only part of the lower hemisphere).

Figure 14

$t=8.1$ top: In the interior of the wormhole $H_R$ (horizontal arrows) is negative (red and inward-directed) everywhere and in the neck center it blows up, while ${}^{(3)}R$ gets smaller since $K^{\widehat{\chi }}$ (tangential arrows) becomes also large but stays positive (green and outward-directed). Any matter inside the neck would get extremely spaghettified: radially stretched by the large positive $K^{\widehat{\chi }}$ but still volume reduced due to the large negative $H_R$. bottom: The blue photon marches upwards. Still no sign of a BH horizon. Compare the latitude of the future trapped region in the bubble of the type II fluctuation.

Figure 15

$t=9.1$ top: The wormhole has been pinched off: a singularity formed that crushed the yellow photon. It is also quite interesting that the singularity formed in a matter free region (between the two black dots). Note that the vertical separation of the cusps in the embedding has no physical meaning. The inner-trapped region in Fig. 10 appeared shortly after horizon entry, suggesting that also in the case of a radiation-filled universe the formation of a noncentral singularity cannot be avoided; pressure forces will try to bring material trough the wormhole, which is effective only when the throat is within the horizon. Once within, it will immediately shrink in the angular direction and stretch in the radial direction. Both processes slow down the pressure driven matter transport. Hence, we expect that any type II fluctuation with $\zeta \ge {\zeta }_{\mathrm{nc}}>{\zeta }_{\max }$ will form a noncentral singularity, where ${\zeta }_{\mathrm{nc}}$ is a new shape-dependetn threshold. It would be interesting to determine ${\zeta }_{\mathrm{nc}}$ by a proper numerical simulation. bottom: The green photon reaches the center.

Figure 16

$t=10.1$ top: The different trapped regions in the baby universe have merged. The blue photon reaches the center. bottom: Comparing the latitude $\chi$ of the lower black dots with the latitude of the BH horizon in the baby universe in the upper panel, we see that the BH horizon will form soon.

Figure 17

$t=11.1$ top: The cut-off “elastic band” has now relaxed: In the baby universe, ${}^{(3)}R$ is everywhere positive. bottom: The BH horizon has now formed around the black mass dots and sealed the fate of the photons and the matter in the overdensity: All the matter and the green photon will be crushed. The blue photon is lucky. (Here Mathematica colored some parts greyish red which actually should be black. Hence the blue photon is really outside the trapped region. The same holds for Fig. 18).

Figure 18

$t=12.1$ top: The orange photon still nearly rests above the horizon. The baby universe is so close to its big crunch that the blue photon will not reach the PBH singularity in the baby universe. bottom: Note that the singularity in the type II fluctuation in Fig. 14 was announced by a blowup of $K$ (while ${}^{(3)}R$ got smaller). Here, it is ${}^{(3)}R$ that blows up while $K$ remains small.

Figure 19

$t=25.1$ Both photons required a long time (twice the age of the universe after PBH formation at $t_{\mathrm{BH}}=12.57$) to finally escape the horizon, although they are still inside the matter free vacuole: they must be strongly redshifted. After $t_{\mathrm{BH}}$, both slices are now physically identical and indistinguishable: a PBH with Schwarzschild radius $R_{\mathrm{BH}}=2.6$ and mass $M_{\mathrm{BH}}=R_{\mathrm{BH}}/2$ inside an expanding vacuole embedded in a flat exact FRW universe. The metric inside the comoving black dots is exactly Schwarzschild such that the existence of the FRW is undetectable using gravity experiments like Kepler law or peri-PBH-shift inside the vacuole. This has been proven by Einstein and Strauss [66].

Figure 20

Conformal diagrams for closed (left) and flat (right) dust FRW.

Figure 21

Spatial shape of $\eta$ in the transition region. Type II is plotted dashed, type I full (shifted for better comparison).

Figure 22

Conformal diagrams of type II (top) and I (bottom) spacetime. The coordinates $\tilde{\eta }$, $\tilde{\chi }$ are chosen to keep $\tilde{\eta }=\eta$, $\tilde{\chi }=\chi$ in the exact closed FRW part and to have a $\tilde{\eta }$-simultaneous big bang and big crunch. The full spacelike lines are $t=\mathrm{const}$ slices and the ones from $t=0.1$ and to$t=12.1$ coincide with the slices shown in the embeddings. Close before the big crunch in the exact closed FRW at $\tilde{\eta }=2\pi$ corresponding to $t=12.56$, we added an additional line at $t=12.55$. Above this in the conformal diagram for the type I fluctuation, the slice at $t=13.1$ terminating in the singularity can be seen, showing that the green photon still lives at this time. The magnified corner of the type II diagram shows the termination of $t>8.15$-slices in the singularity coming from the flat FRW. They reappear from the singularity well visibly on the left. This effect corresponds to the seeming topology change we have seen in the embedding diagrams of the type II fluctuation. For the type I fluctuation spatial slices coming from the flat FRW only end in the singularity but do not reappear for smaller $\tilde{\chi }$. If we had used the $\tilde{\eta }$-slicing in the embeddings, the evolution of type II would have looked very similar to the one of type I. Hence the type I and II spacetimes are very similar. The two dashed timelike lines correspond to the comoving boundaries that separate the exact closed and flat FRW parts from the matching region. They are depicted as black dots in the embeddings. The straight, diagonal (colored) null lines correspond to the large (colored) photon dots in the embeddings. Note that in the type I diagram the upper part of the dark (blue) photon line is approximately the BH event horizon, while for the type II spacetime it is a photon line starting in the left corner which is approximately the orange photon.