Primordial black holes from an interrupted phase transition

Technical Summary: Primordial Black Holes from an Interrupted Phase Transition

Problem Statement
Primordial Black Holes (PBHs) are compelling candidates for dark matter and potential explanations for various cosmological anomalies, yet their formation mechanisms remain an open question. While standard models rely on the gravitational collapse of large curvature perturbations generated during inflation (requiring specific potential features like inflection points or plateaus), or on phase transitions occurring during cooling, this paper addresses the possibility of PBH formation during the reheating epoch. Specifically, the authors investigate a scenario where a first-order phase transition (FOPT) is initiated by the rising temperature of the radiation bath during reheating but is subsequently "interrupted" before completion.

Methodology and Setup
The authors propose a mechanism occurring in the early matter-dominated stage of reheating, driven by the decay of a pressureless fluid (the "reheaton," $\chi$) into a relativistic plasma. The temperature of this plasma, $T$, initially rises to a maximum value $T_{\text{max}}$ before decreasing as the Universe expands.

The core of the mechanism involves a real scalar field $\phi$ undergoing a symmetry-restoring FOPT as the temperature increases from zero to $T_{\text{max}}$. The scenario is defined by a specific hierarchy of temperatures:

1. $T_c$: The critical temperature where the symmetry-breaking and symmetry-restoring vacua are degenerate.
2. $T_n$: The nucleation temperature required for the phase transition to complete (bubble nucleation rate overcoming expansion).
3. $T_1$: The spinodal temperature where the potential barrier vanishes.

The "interrupted" nature of the transition arises from the condition $T_c < T_{\text{max}} < T_n \lesssim T_1$. In this regime, the temperature rises high enough to trigger bubble nucleation of the symmetry-restoring phase but not high enough for the bubbles to percolate and complete the transition before the temperature peaks and begins to fall.

Key Contributions and Mechanism
The paper details the fate of bubbles nucleated under these conditions:

• Expansion and Contraction: Bubbles nucleate around $T_{\text{max}}$ and expand while $T > T_c$. As the temperature drops back below $T_c$, the free energy difference becomes negative, causing the bubbles to shrink and eventually disappear at a scale factor $a_{\text{zero}}$.
• Density Perturbations: The expansion and subsequent contraction of the bubble walls transfer energy between vacuum and thermal forms. This process leaves behind a macroscopic, spherically symmetric region with a positive density perturbation ($\delta_i$). The authors derive an expression for this initial density contrast, $\delta_i$, which depends on the vacuum energy difference $|\Delta V_0|$ and the ratio of scale factors $a_{c,2}/a_{\text{max}}$ (where $a_{c,2}$ is the scale factor when the bubble stops expanding).
• PBH Formation via Accretion: Unlike standard collapse scenarios, these perturbations do not immediately collapse. Instead, they act as seeds for the "post-collapse accretion mechanism" during the matter-dominated era. The overdense region accretes surrounding reheatons, leading to a non-linear growth of the density contrast. This eventually triggers the collapse of the entire region into a PBH.

Results and Abundance Estimation

• PBH Mass: The final mass of the PBH is determined primarily by the reheating temperature ($T_{\text{RH}}$) rather than the specific details of the phase transition, as the mass grows until radiation domination begins. The estimated mass is given by $M_{\text{PBH}} \sim 3.5 \times 10^{-12} M_\odot \alpha (10^5 \text{ GeV}/T_{\text{RH}})^2$, suggesting a monochromatic distribution.
• Abundance: The relic abundance ($f_{\text{PBH}}$) is estimated by counting the number of bubble nucleations around $T_{\text{max}}$. The calculation depends on the bubble nucleation rate $\Gamma(T)$, parametrized by the rapidity parameter $\hat{\beta}_{\text{max}} = -d(S_3/T)/d\ln T |_{T_{\text{max}}}$.
• Phenomenological Viability: Using benchmark values ($\hat{\beta}_{\text{max}} \sim 10^5$, $a_{\text{RH}}/a_{\text{max}} \sim 10$), the authors demonstrate that the required nucleation rate to produce a significant PBH abundance (potentially constituting all dark matter) is consistent with current observational constraints from Big Bang Nucleosynthesis (BBN), Cosmic Microwave Background (CMB) anisotropies, microlensing, and gravitational wave limits.

Significance and Claims
The paper claims to propose a novel PBH formation mechanism that does not rely on inflationary curvature perturbations or standard cooling phase transitions. Instead, it utilizes the unique thermal history of reheating to create an "interrupted" phase transition. The authors argue that this scenario naturally generates macroscopic overdensities that can collapse into PBHs via accretion.

The significance lies in the fact that this mechanism can produce a substantial abundance of PBHs with a mass spectrum determined solely by the reheating temperature, making it a testable prediction for future PBH searches. The authors note that while the scenario relies on assumptions regarding the sphericity of the bubbles and the homogeneity of the reheaton fluid, these are standard assumptions in PBH formation literature. They acknowledge that quantitative investigations into potential instabilities (e.g., at the turnaround point) and the effects of initial inhomogeneities are left for future work. The mechanism is shown to be robust across a wide range of parameters, including those derived from a benchmark Abelian Higgs model.