Dynamical backreaction of a mass-acquiring scalar field on first-order phase transitions

TL;DR

This work demonstrates that a mass-acquiring spectator field $\chi$ coupled to the field driving a first-order phase transition can backreact dynamically on bubble expansion even when the nucleation potential is fixed. Through lattice simulations, the authors uncover a two-fold effect: friction-like energy transfer to $\chi$ during wall propagation and, more prominently, a rapid suppression of $\chi$ oscillations inside true-vacuum bubbles that lowers the effective true-vacuum minimum and releases additional vacuum energy, accelerating the transition and enhancing the gravitational-wave signal. They develop an analytical framework linking the GW amplitude to the net released vacuum energy after accounting for energy transfer to $\chi$, including a semi-analytic expression for the transferred energy $\Delta E_{\chi}$ and a scaling $\Omega_{\mathrm{GW}}^{\text{peak}}\propto(\rho_{\text{vac}}-\Delta E_{\chi})^2$, enabling improved GW predictions for related phase-transition scenarios. The results have broad implications for early-Universe cosmology, particularly in models with mass-generating spectator fields, where standard estimates based solely on the initial potential may underestimate the GW signal.

Abstract

Phase transitions in the early Universe give rise to effective masses for massless fields in the symmetry-broken phase. We perform lattice simulations to study the dynamical impact of a mass-acquiring spectator field on the evolution of first-order phase transitions and the associated gravitational-wave production, while keeping the effective potential responsible for bubble nucleation fixed. In addition to the well-known friction effects, we identify a novel effect that significantly enhances the strength of first-order phase transitions. In contrast to the general scenario, although the effective potential governs the tunneling rate, the amplitude of the $χ$ field is strongly suppressed inside the true vacuum bubble, resulting in a faster bubble expansion than predicted by the effective potential alone. The amplitude of the mass-acquiring field is highly suppressed in the true vacuum bubbles, resulting in additional release of vacuum energy that concentrate on the bubble walls. We further develop an analytical framework that not only explains our numerical results but can also be used to improve the estimation of gravitational-wave signals in related phase-transition scenarios.

Figures (10)

• Figure 1: Schematic comparison between the effective potential without the interaction term $V(\phi)$ (blue) and the reduced potential $V_{\text{re}}(\phi)$ (red), where the latter is fixed in our subsequent analysis. The potential energy density at the false vacuum is labeled $V_0$ and the vacuum expectation values in the broken phase are denoted $\phi_{\text{b}}$ and $\eta$ for $V(\phi)$ and $V_{\text{re}}(\phi)$, respectively. The corresponding vacuum energy differences are labeled $\rho_{\text{vac}}$ and $\tilde{\rho}_{\text{vac}}$, respectively.
• Figure 2: Lorentz factor $\gamma$ of bubble walls as a function of normalized radius $R/R_c$. In both panels, the black dashed line represents the analytical prediction $\gamma = R/R_c$ from the thin-wall approximation in the absence of coupling. Note that the parameters of light blue curve in the right panel does not appear in the multi-bubble simulations.
• Figure 3: The evolution of the scalar field $\phi(x)/\eta$ in the absence of coupling ($\lambda_{\phi\chi} = 0$ and $A=0$). From left to right, each panel corresponds to snapshots at $t\omega_* = 10$, 20, 30, and 40, respectively.
• Figure 4: The evolution of $\phi$ and $\chi$ and the energy distribution with $\lambda_{\phi\chi} = 5\lambda_\phi$ and $A = 1$. The top row shows the field configuration of $\phi(\mathbf{x})/\eta$, the middle row displays $\chi^2(x,y) / \eta^2$, and the bottom row presents the total energy density of $\chi$, namely $E_\chi = V_{\phi\chi} + K_\chi + G_\chi$, which is normalized by $\omega_*^2 \eta^2$. From left to right, each column corresponds to snapshots at $t\omega_* = 10$, 20, 30, and 40, respectively.
• Figure 5: This figure shows the evolution of the energy fraction of each components, normalized to the total energy density $\rho_{\text{tot}}$, with the parameters $\lambda_{\phi\chi} = 5\lambda_\phi$ and $A = 1$. The components include kinetic ($K$) and gradient ($G$) energies of $\phi$ and $\chi$, interaction energy $V_{\phi\chi}$, and the shifted potential energy of $\phi$, $V(\phi) - V(\phi_\mathrm{b})$. The black dashed line represents the sum of these components, which remains constant throughout the simulation, thereby verifying the numerical accuracy.