Primordial Black Hole Formation during First-Order Phase Transitions

TL;DR

Problem: The formation of primordial black holes (PBHs) in the early universe may be enhanced if horizon-entry fluctuations occur during a first-order phase transition. Approach: The authors perform general-relativistic hydrodynamics simulations with a bag-model equation of state describing high- and low-energy density phases, a latent heat $L$, and a mixed phase with $v_s^{eff}\approx 0$, tracking horizon-scale overdensities through PBH formation. Findings: The simulations reveal a two-zone density structure with a high-energy-density core in a mixed-phase environment, yielding PBHs with masses tied to the horizon mass $M_h$ and showing that the formation threshold $\delta_c^{FPT}$ can be substantially below the ordinary RD value; increasing transition strength $\eta$ lowers the threshold, and a corrected threshold exhibits a double-dip feature, though a simple critical scaling exponent $\gamma$ is not clearly established. Significance: If robust, phase-transition–driven PBH production would bias PBH masses toward horizon-scale values, impacting early-universe cosmology, dark matter considerations, and gravitational-wave expectations, and motivates higher-resolution simulations to map the mass spectrum and scaling behavior more precisely.

Abstract

Primordial black holes (PBHs) may form in the early universe when pre-existing adiabatic density fluctuations enter into the cosmological horizon and recollapse. It has been suggested that PBH formation may be facilitated when fluctuations enter into the horizon during a strongly first-order phase transition which proceeds in approximate equilibrium. We employ general-relativistic hydrodynamics numerical simulations in order to follow the collapse of density fluctuations during first-order phase transitions. We find that during late stages of the collapse fluctuations separate into two regimes, an inner part existing exclusively in the high-energy density phase with energy density $ε_{\rm h}$, surrounded by an outer part which exists exclusively in the low-energy density phase with energy density $ε_{\rm h}-L$, where $L$ is the latent heat of the transition. We confirm that the fluctuation density threshold $δε/ε$ required for the formation of PBHs during first-order transitions decreases with increasing $L$ and falls below that for PBH formation during ordinary radiation dominated epochs. Our results imply that, in case PBHs form at all in the early universe, their mass spectrum is likely dominated by the approximate horizon masses during epochs when the universe undergoes phase transitions.

Figures (4)

• Figure 1: Energy density, $\epsilon$, as a function of scaled circumferential radius, $R_{\rm sc}=(R/R_{\rm h}(t_0))(a_0/a)$, for a fluctuation with initial density contrast, $(\delta\epsilon /\epsilon)_{\rm hc}= 0.535$, at horizon crossing. The initial horizon at $t_0$ is located at $R_{\rm sc}=1$. From top to bottom, solid lines show the fluctuation at 1., 1.22, 1.49, 1.82, 2.23, 2.72, 3.32, 4.06, 4.95, 6.05, 7.39, 9.03, 11.0, 13.5, 16.4, and 20.1 times the initial time $t_0$. Constant proper time slicing was used. The horizontal dashed lines indicate the energy densities at onset and completion of the phase transition. The formation of a PBH with $M_{\rm pbh}\approx 0.34M_{\rm h}(t_0)$ results.
• Figure 2: A zoom into the central region of Figure 1. From top to bottom, solid lines show the fluctuation at 1.0, 1.22, 1.49, 1.82, 2.22, 2.72, 3.32, 4.06, 4.95, and 5.47 times the initial time $t_0$. The horizontal dashed lines indicate the energy densities at onset and completion of the phase transition. The dotted lines shows, for comparison, the evolution of a fluctuation with the same initial fluctuation parameters, but entering the cosmological horizon during an epoch with equation of state $p =\epsilon /3$.
• Figure 3: Coordinate velocity, $\partial R/\partial\tau$, with $\tau$ proper time of fluid elements, as a function of scaled circumferential radius, $R_{\rm sc}=(R/R_{\rm h}(t_0))(a_0/a)$. The solid lines show, from top to bottom, the coordinate velocities at times 1.0, 1.22, 1.49, 1.82, 2.22, 2.72, 3.32, 4.06, and 4.95 times $t_0$ for the same fluctuation and equation of state as shown in Figures 1 and 2 by the solid lines.
• Figure 4: Energy overdensity threshold for PBH formation, $\delta_{\rm c}^{\rm FPT}$ (solid line), for fluctuations entering the cosmological horizon during, or close to a first-order phase transition, as a function of horizon crossing time, $\tau_{\rm hc}=\epsilon_0(t_0) /\epsilon_{\rm h}(T_{\rm c})$. The energy densities at the onset and completion of the transition are chosen $\epsilon_{\rm h}(T_{\rm c}) =1$ and $\epsilon_{\rm l}(T_{\rm c}) =0.5$, respectively. The crosses are points determined from numerical simulation. The solid line is an interpolation between crosses. The dotted line shows $\delta_{\rm c}^{\rm FPT}/(1+w)$, with $w$ the cosmic average $p/\epsilon$ at horizon crossing of the fluctuation. (See text for further explanations).