Numerical simulations of primordial black hole formation via delayed first-order phase transitions

Abstract

We perform fully nonlinear, spherically symmetric numerical simulations of superhorizon false-vacuum-domain (FVD) collapse in a coupled gravity-scalar-fluid system to study primordial black hole (PBH) formation during delayed first-order phase transitions (FOPTs). Using adaptive mesh refinement to resolve the bubble wall, we identify three dynamical outcomes: type B (supercritical) PBHs with an interior baby universe and a bifurcating trapping horizon, type A (subcritical) PBHs with an apparent horizon formed by direct wall collapse, and dispersal with no PBH formation. To separate these three cases, we evaluate two commonly used PBH-formation criteria: the time scale ratio $t_\mathrm{H}/t_\mathrm{V}$ (horizon crossing time versus vacuum-energy domination time) and the local density contrast $δ(t_\mathrm{H})$ at horizon crossing. For the parameter space explored, we find that $t_\mathrm{H}/t_\mathrm{V}$ is a more robust predictor of outcome: type B PBHs form when $t_\mathrm{H}/t_\mathrm{V} \gtrsim 1$ (critical range $\sim 1.1 - 1.6$ in our survey), type A PBHs arise when $t_\mathrm{H}/t_\mathrm{V}$ is below this threshold but remains above a lower bound (typical range $\sim 0.35 - 0.7$), and no-PBH dispersal occurs when $t_\mathrm{H}/t_\mathrm{V}$ falls below this lower bound. When a clear thin-wall FVD boundary exists, $δ(t_\mathrm{H})$ can correspondingly distinguish different outcomes (roughly $δ_c(t_\mathrm{H}) \sim 1 - 1.7$ for type B and $δ_c(t_\mathrm{H}) \sim 0.35 - 0.5$ for type A), but is highly sensitive to wall structure and model details and thus less universal. These results offer new insights into the dynamics of FVD collapse, quantify practical PBH-formation thresholds, and pave the way for precise predictions of PBH abundance from delayed FOPTs.

Figures (8)

• Figure 1: The scalar potential $V(\phi)$ for different values of $\lambda$. The potential is normalized by $V_\mathrm{F}$ and the field by $\phi_\mathrm{T}$.
• Figure 2: Time evolution of several quantities for type B PBH formation with $V_\mathrm{F} = 5\times10^{-5}$. (a) Scalar-field profile $\phi$. (b) Areal radius $R$ before the formation of the bifurcating trapping horizon. (c) Total energy density $\rho_\mathrm{tot}$. (d) Expansions $\Theta^\pm$; the black dot marks the bifurcating trapping horizon. (e) Areal radius $R$ after horizon formation. (f) Three-dimensional Ricci scalar ${}^{(3)}R$.
• Figure 3: Time evolution of several quantities for type A PBH formation with $V_\mathrm{F} = 1\times10^{-5}$. (a) Scalar-field profile $\phi$. (b) Areal radius $R$ before the formation of the apparent horizon. (c) Total energy density $\rho_\mathrm{tot}$. (d) Expansions $\Theta^\pm$; the black dot marks the apparent horizon. (e) Areal radius $R$ after horizon formation. (f) Three-dimensional Ricci scalar ${}^{(3)}R$.
• Figure 4: Time evolution of several quantities for the no-PBH outcome with $V_\mathrm{F} = 5\times10^{-7}$. (a) Scalar-field profile $\phi$. (b) Areal radius $R$. (c) Total energy density $\rho_\mathrm{tot}$. (d) Expansions $\Theta^\pm$. (e) Three-dimensional Ricci scalar ${}^{(3)}R$.
• Figure 5: Thresholds for type B PBH formation. (a) Dependence on the initial FVD radius $r_i$ (fixed $\phi_\mathrm{T} = 0.1$ and $\lambda = 20$). (b) Dependence on the potential parameter $\lambda$ (fixed $\phi_\mathrm{T} = 0.1$ and $r_i = 10$). (c) Dependence on the true-vacuum field value $\phi_\mathrm{T}$ (fixed $\lambda = 20$ and $r_i = 10$). Blue squares and orange circles denote the critical values of $t_\mathrm{H}/t_\mathrm{V}$ and $\delta(t_\mathrm{H})$, respectively.