Heterogeneous Cosmological Phase Transitions: Seeded by Domain Walls and Junctions

TL;DR

This work develops and applies a comprehensive framework for heterogeneous cosmological phase transitions seeded by domain walls and their junctions. By modeling defect-seeded bubbles as spherical caps and using both thin-wall and tanh-profile approximations alongside numerical MPT solutions, the authors show that wall- and junction-seeded channels can complete a first-order transition at higher temperatures than homogeneous nucleation, with junctions often providing the most efficient seed. The analysis spans Z2 and Z_n (n≥3) symmetric two-field models, revealing that domain-wall networks and Y/X-type junctions modify nucleation and percolation dynamics, with clear implications for gravitational-wave signals and electroweak baryogenesis. Overall, defect-induced nucleation opens a robust, testable channel in early-universe cosmology, warranting further study of network dynamics and observational consequences.

Abstract

Heterogeneous nucleation is central to many familiar first-order phase transitions such as the freezing of water and the solidification of metals, and it can also play a crucial role in cosmology. We examine nucleation seeded by preexisting domain walls and demonstrate its strong impact on the dynamics of cosmological phase transitions. The bubble solutions take the form of spherical caps, and the contact angle is fixed by the ratio of the domain-wall tension to the bubble-wall tension. A larger domain-wall tension, or equivalently a smaller contact angle, reduces the wall-seeded bubble volume and lowers the critical nucleation action. For theories with $\mathbb{Z}_{n\geq 3}$ symmetry, domain-wall junctions naturally appear and we find that they seed nucleation even more efficiently than the walls themselves. Using a two-scalar-field model as an illustration, we compute nucleation temperatures for both homogeneous and heterogeneous channels and show that junction-seeded nucleation occurs at a higher temperature and is the dominant mechanism that completes the first-order cosmological phase transition.

Figures (10)

• Figure 1: Left panel: 3d visualization of a nucleated bubble attached to the domain wall (side view). Middle panel: top view. Right panel: notation for the top view, showing the orange spherical cap as the bubble wall and the purple straight line as the domain wall. The purple dashed line indicates the disappearance ("melting") of the domain wall inside the nucleated bubble. The contact angle is defined as $\theta$, $R_\parallel = R \sin{\theta}$ and $R_\perp = R(1-\cos{\theta})$.
• Figure 2: Illustration of a type-I X-type junction decaying into a linked Y-junction configuration when $\sigma_{\rm DW,2} < \sqrt{2}\,\sigma_{\rm DW,1}$. The thicker line in the right panel indicates a larger wall tension.
• Figure 3: Illustration of the geometric shape of the spherical-cap bubble, shown in orange, together with the domain wall, shown in purple. (a) The 3d shape of the bubble on the $\mathbb{Z}_3$ junction, where the angle between adjacent walls is $2\pi/3$. To maintain continuity of the bubble across the wall, the center of the sphere is constrained to lie along the outer angle bisector of the two walls. (b) The $z=0$ slice of the bubble at the junction. (c) Notation for the $z=0$ slice.
• Figure 4: Ratios of the critical bubble actions for the wall-induced, Y-type–junction–induced, and X-type–junction–induced cases relative to the homogeneous spherical bubble, based on the thin-wall approximation. The orange Y-type junction curve has an upper bound at $\cos\theta = \sqrt{3}/2$, while the green X-type junction curve has an upper bound at $\cos\theta = 1/\sqrt{2}$. The horizontal blue dashed line is at approximately $3/4$ and intersects the blue curve at $\cos\theta = 0.16$. The horizontal orange dashed line is at approximately $1/2$ and intersects the orange curve at $\cos\theta = 0.24$ and the green curve at $\cos\theta = 0.19$.
• Figure 5: Left panel: The action $S_3/T$ as a function of $T$ for the model with $\mathbb{Z}_2$ domain walls. The model parameters are $\kappa=1.3$, $\lambda = 1.6$, $\eta = 0.129$, $\mu_\phi=88$ GeV and $\mu_S = 127$ GeV. For the wall-seeded actions, we include the results from the exact solution based on the mountain-pass theorem (MPT), the thin-wall approximation, and the tanh-profile approximation. For this model point, $T_c = 109~\text{GeV}$ and $\cos\theta = \sigma_{\rm DW}/(2\sigma_{\rm B}) = 0.72$ as $T \to T_c$. The horizontal dashed lines indicate the critical nucleation actions, with $S^{(3)}_{3,c}/T = 144$ for homogeneous nucleation and $S^{(3)}_{3,c}/T = 109$ for wall-seeded nucleation. Right panel: Same as the left panel, but zoomed in on the region with $T$ close to $T_c$.