## Author(s)

Cai, Rong-Gen, Hao, Yu-Shi, Wang, Shao-Jiang## Abstract

Recently, much attention has been focused on the false-vacuum islands that are flooded by an expanding ocean of true-vacuum bubbles slightly later than most of the other parts of the world. These delayed decay regions will accumulate locally larger vacuum energy density by staying in the false vacuum longer than those already transited into the true vacuum. A false-vacuum island with thus acquired density contrast of a super-horizon size will evolve locally from radiation dominance to vacuum dominance, creating a local baby Universe that can be regarded effectively as a local closed Universe. If such density contrasts of super-horizon sizes can ever grow large enough to exceed the threshold of gravitational collapse, primordial black holes will form similar to those collapsing curvature perturbations on super-horizon scales induced by small-scale enhancements during inflation. If not, such density contrasts can still induce curvature perturbations potentially observable today. In this paper, we revisit and elaborate on the generations of primordial black holes and curvature perturbations from delayed-decayed false-vacuum islands during asynchronous first-order phase transitions with fitting formulas convenient for future model-independent studies.

## Figures

A schematic illustration for the delayed-decayed false-vacuum regions to accumulate (false) vacuum energy densities that gradually dominate the evolution of these overdensities, slightly shrinking the local comoving Hubble horizon that effectively induces local close patches. The subsequent bubble nucleations within these local baby Universes would prevent them from entering into a state of eternal inflation and hence being detached from our Universe, ensuring the conditions for PBH formations.

A schematic illustration for the delayed-decayed false-vacuum regions to accumulate (false) vacuum energy densities that gradually dominate the evolution of these overdensities, slightly shrinking the local comoving Hubble horizon that effectively induces local close patches. The subsequent bubble nucleations within these local baby Universes would prevent them from entering into a state of eternal inflation and hence being detached from our Universe, ensuring the conditions for PBH formations.

A schematic illustration for the delayed-decayed false-vacuum regions to accumulate (false) vacuum energy densities that gradually dominate the evolution of these overdensities, slightly shrinking the local comoving Hubble horizon that effectively induces local close patches. The subsequent bubble nucleations within these local baby Universes would prevent them from entering into a state of eternal inflation and hence being detached from our Universe, ensuring the conditions for PBH formations.

\textit{Left}: The time evolution of the false-vacuum fraction given earlier (solid) and later (dashed) nucleation times of the first bubble with shorter (red) and longer (blue) duration times. \textit{Right}: The physical (phenomenological) strength and duration parameters are bounded in the region with $0.1<\alpha_*<1.0$ and $1<(\beta/H)_*<20$ (dashed) given the model parameters $\bar{\alpha}$ and $\bar{\beta}$. The approximation $H_\mathrm{nuc}t_\mathrm{nuc}\approx1/2$ is checked with solid contour curves.

\textit{Left}: The time evolution of the false-vacuum fraction given earlier (solid) and later (dashed) nucleation times of the first bubble with shorter (red) and longer (blue) duration times. \textit{Right}: The physical (phenomenological) strength and duration parameters are bounded in the region with $0.1<\alpha_*<1.0$ and $1<(\beta/H)_*<20$ (dashed) given the model parameters $\bar{\alpha}$ and $\bar{\beta}$. The approximation $H_\mathrm{nuc}t_\mathrm{nuc}\approx1/2$ is checked with solid contour curves.

The time evolutions of energy densities with respect to the total density at the initial nucleation time (top) and the real time (bottom) for radiations (red) and vacuum (blue) components as well as their combined (black) in the normal-decayed (solid) and delayed-decayed (dashed) regions during an illustrative FOPT with a strength parameter $\alpha_*=0.5$ and a duration parameter $(\beta/H)_*=10$. PBHs can be formed at the latest at the gray vertical line (right) for a delayed-decayed nucleation time at the earliest at the left boundary of the green band, below which no PBH can be produced (left).

The time evolutions of energy densities with respect to the total density at the initial nucleation time (top) and the real time (bottom) for radiations (red) and vacuum (blue) components as well as their combined (black) in the normal-decayed (solid) and delayed-decayed (dashed) regions during an illustrative FOPT with a strength parameter $\alpha_*=0.5$ and a duration parameter $(\beta/H)_*=10$. PBHs can be formed at the latest at the gray vertical line (right) for a delayed-decayed nucleation time at the earliest at the left boundary of the green band, below which no PBH can be produced (left).

The time evolutions of energy densities with respect to the total density at the initial nucleation time (top) and the real time (bottom) for radiations (red) and vacuum (blue) components as well as their combined (black) in the normal-decayed (solid) and delayed-decayed (dashed) regions during an illustrative FOPT with a strength parameter $\alpha_*=0.5$ and a duration parameter $(\beta/H)_*=10$. PBHs can be formed at the latest at the gray vertical line (right) for a delayed-decayed nucleation time at the earliest at the left boundary of the green band, below which no PBH can be produced (left).

The time evolution of the overdensity (left) and its zoom-in view (right) in the delayed-decayed region during a FOPT with illustrative strength factor $\alpha_*=0.5$ and inverse duration $(\beta/H)_*=10$, where vertical dashed lines are different delayed-decayed nucleation times of the first bubble with possible PBH formation at later times as shown with vertical solid lines.

The time evolution of the overdensity (left) and its zoom-in view (right) in the delayed-decayed region during a FOPT with illustrative strength factor $\alpha_*=0.5$ and inverse duration $(\beta/H)_*=10$, where vertical dashed lines are different delayed-decayed nucleation times of the first bubble with possible PBH formation at later times as shown with vertical solid lines.

The fitting formulas (dashed) to the PBH mass (left) and abundance (right) given the FOPT parameters from the strength factor $\alpha_*$, inverse duration $(\beta/H)_*$, and percolation temperature $T_*$ for the most probable case of PBH formations.

The fitting formulas (dashed) to the PBH mass (left) and abundance (right) given the FOPT parameters from the strength factor $\alpha_*$, inverse duration $(\beta/H)_*$, and percolation temperature $T_*$ for the most probable case of PBH formations.

Illustrative examples of FOPTs at different percolation temperatures $T_*$ with different strength factors $\alpha_*$ and inverse durations $(\beta/H)_*=4$ (vertical dashed lines) and $(\beta/H)_*=3$ (vertical solide lines), in which cases the associate PBH masses and abundances roughly reach the boundaries of current PBH constraints from the collection~\cite{Carr:2020xqk} (and references therein but with GW constraints replaced by recent updates.~\cite{Hutsi:2020sol,Nitz:2022ltl,Chen:2021nxo}).

The standard deviation (top) and dimensionless power spectrum (bottom) for the accumulated overdensity perturbations filtered at a scale that enters the Hubble horizon around the bubble percolation time. The parameter dependences are shown separately with respect to the strength factor $\alpha_*$ (left) and inverse duration $(\beta/H)_*$ (right) for some fixed values of $(\beta/H)_*$ and $\alpha_*$, respectively.

The standard deviation (top) and dimensionless power spectrum (bottom) for the accumulated overdensity perturbations filtered at a scale that enters the Hubble horizon around the bubble percolation time. The parameter dependences are shown separately with respect to the strength factor $\alpha_*$ (left) and inverse duration $(\beta/H)_*$ (right) for some fixed values of $(\beta/H)_*$ and $\alpha_*$, respectively.

The standard deviation (top) and dimensionless power spectrum (bottom) for the accumulated overdensity perturbations filtered at a scale that enters the Hubble horizon around the bubble percolation time. The parameter dependences are shown separately with respect to the strength factor $\alpha_*$ (left) and inverse duration $(\beta/H)_*$ (right) for some fixed values of $(\beta/H)_*$ and $\alpha_*$, respectively.

The power spectra for the accumulated overdensities (top panel) with fitting formual shown in white dashed contours and the induced curvature perturbations (bottom panels) with corresponding ranges of percolation temperatures to manifest the observational exclusion constraints (white contours).

The power spectra for the accumulated overdensities (top panel) with fitting formual shown in white dashed contours and the induced curvature perturbations (bottom panels) with corresponding ranges of percolation temperatures to manifest the observational exclusion constraints (white contours).

The power spectra for the accumulated overdensities (top panel) with fitting formual shown in white dashed contours and the induced curvature perturbations (bottom panels) with corresponding ranges of percolation temperatures to manifest the observational exclusion constraints (white contours).

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