Primordial black hole formation by vacuum bubbles

TL;DR

The paper investigates primordial black hole formation from vacuum bubbles nucleated during inflation, employing numerical GR+radiation simulations to track bubble dynamics and horizon formation. It demonstrates two distinct outcomes: subcritical bubbles collapsing into black holes with masses set by the initial conserved bubble mass, and supercritical bubbles inflating into baby universes that form wormholes, yielding two black holes with masses that approach a bound $M_{bh} \,\sim M_{Pl}^2 H_i R_i^2$ for large bubble radii. The resulting black hole mass spectrum is extremely broad, governed by the inflationary bubble size distribution, with a transition scale $M_*$ separating a shallow $M^{-1/2}$ tail from a flat regime, and upper/lower cutoffs $M_{min}$ and $M_H/M_F$. The authors discuss observational implications, showing that PBHs in this scenario could contribute up to ~10% of dark matter and could account for LIGO mass range events or serve as seeds for supermassive black holes under suitable nucleation rates, while remaining compatible with current constraints on extended mass functions.

Abstract

Vacuum bubbles may nucleate during the inflationary epoch and expand, reaching relativistic speeds. After inflation ends, the bubbles are quickly slowed down, transferring their momentum to a shock wave that propagates outwards in the radiation background. The ultimate fate of the bubble depends on its size. Bubbles smaller than certain critical size collapse to ordinary black holes, while in the supercritical case the bubble interior inflates, forming a baby universe, which is connected to the exterior region by a wormhole. The wormhole then closes up, turning into two black holes at its two mouths. We use numerical simulations to find the masses of black holes formed in this scenario, both in subcritical and supercritical regime. The resulting mass spectrum is extremely broad, ranging over many orders of magnitude. For some parameter values, these black holes can serve as seeds for supermassive black holes and may account for LIGO observations.

Figures (13)

• Figure 1: A simple example of a two-field potential where the bubble nucleation scenario we discuss in this paper would be possible. As the inflaton field slowly rolls towards our vacuum, it can tunnel through a barrier to another vacuum, which will be the bubble interior.
• Figure 2: A conformal diagram showing the formation of a black hole by a subcritical bubble in the background of a radiation dominated spatially flat FRW universe. At the time $t_{i}$, when inflation ends, the bubble (the shaded region of the diagram) expands with a large Lorentz factor relative to the Hubble flow. The bubble wall is represented by a thick blue solid curve. The bubble expansion is slowed down by momentum transfer to the ambient radiation, and eventually the bubble turns around and collapses into a Schwarzschild singularity (red solid line). The thick dashed curve represents the shock front propagating at the speed of sound, caused by the impact of the fast-moving wall on the radiation. Region outside the shock front is an unperturbed FRW universe. The spacelike curve below the Schwarzschild singularity is the black hole apparent horizon, which is used to represent the black hole boundary in our simulations. It lies inside the event horizon (thin dashed straight line).
• Figure 3: A conformal diagram showing the formation of a black hole by a supercritical bubble in a radiation dominated flat FRW universe. In this case, the bubble does not collapse into a singularity. Instead, it grows exponentially in a baby universe, which is connected by a wormhole to the parent FRW universe. The thick dashed curve represents the shock front propagating at the speed of sound, caused by the interaction of the fast-moving wall and the radiation. Region outside the shock front is FRW dominated by homogeneous radiation. The two intersecting spacelike curves below the Schwarzschild singularity are the apparent horizons. The parts above the intersection are black hole apparent horizons, representing the boundary of two black holes. The right branch below the intersection goes lightlike as it approaches the FRW lightlike infinity. This null line is the Hubble radius (or cosmological apparent horizon) of the FRW universe. The two intersecting thin dashed straight lines below the apparent horizons are the event horizons.
• Figure 4: The initial profiles of $B$ and $\dot{R}/R$ for a bubble with $R_{i}=4.$
• Figure 5: The radiation energy density $\rho$ as a function of the comoving radius $r$ at different moments of time outside of a subcritical bubble with $H_{\rm{b}}=0.05H_{i}$, $H_{\sigma}\approx0.03H_{i},$ and $R_{i}=5H_{i}^{-1}$. For all moments, $\rho$ has been rescaled so that the FRW density is 1. (a), (b) and (c) are taken at FRW times $t$ when $t-t_{i}\ll t_{i}$, while (d) is at $t$ when $t-t_{i}\sim t_{i}$. (a) An overdense layer is formed next to the wall as it hits the fluid. (b) A shock wave forms and propagates outwards, while the density at the wall begins to decrease. (c) The shock continues to propagate with the density contrast across the shock rapidly decreasing. (d) $\rho$ right outside the wall becomes much smaller that the FRW density, as if the bubble is surrounded by an empty layer, much like in the case of a dust background.