Slow-down of expanding bubbles in the early Universe

Abstract

We study slow-down effects for bubbles formed in a cosmological first-order phase transition (PT) focusing on deflagrations and hybrids, where the bubble wall is preceded by a shockwave of heated plasma. Slow-down has been observed in multi-bubble simulations together with a suppression of gravitational wave (GW) emission, mostly for slow walls. We study the impact of the shock waves on the wall velocity around percolation, by considering steady-state single-bubble solutions and incorporating the possible heating effects by two different mechanisms. First, we investigate the slow-down experienced by a bubble expanding into an impeding shockwave, where the temperature is higher than at nucleation, and the fluid is no longer at rest. Taking into account such heating and kinematic effects, we find that the most significant slow-down occurs for the fastest walls, and thus cannot explain the suppression of the GWs observed in the simulations. However, these effects are stronger for PTs with a sizeable change in degrees of freedom unlike what is usually implemented in simulations, suggesting that the degrees of freedom can be an important additional parameter for characterizing the GW spectrum. For the second slow-down mechanism, we study heated droplets of false vacuum that shrink towards the end of the PT. By implementing a suitable boundary condition motivated by energy conservation, we show how the droplet velocity, interpreted here as the late-time velocity of the bubble walls, can be predicted from the properties of the initial deflagration/hybrid, in remarkable agreement with numerical simulations. Droplets are found to shrink more slowly for stronger PTs and slower deflagrations, with mild dependence on the change of degrees of freedom. Such slow droplets naturally correlate with a suppression of GWs, while geometrical properties such as the shock width play an important role as well.

Figures (14)

• Figure 1: Field profile $\phi(z)$ for one of the solutions of the coupled plasma-wall system.
• Figure 2: Temperature and velocity profile obtained from energy-momentum conservation. The asymptotic values of these quantities both in front and behind the wall obtained from the matching conditions are also plotted.
• Figure 3: Plot of the friction coefficient $\eta$ in blue and $\tilde{\eta}$ in orange as a function of the wall velocity $\xi_w$. The left panel shows the values for the SM with a low-cutoff with $\alpha_N = 0.021$, corresponding to $\Lambda = 650$ GeV. This model has a local thermal equilibrium solution represented in the figure by the purple star that is consistently found as the $\eta \to 0$ limit in our procedure to solve the Klein-Gordon equation of motion. The right panel shows the friction coefficients for the Bag Model with $\delta a/a = 5.9 \times 10^{-3}$ and $\alpha_N = 0.11$.
• Figure 4: Heating and kinematic effects on the value of the bubble wall velocity $\tilde{\xi}_w$ as a function of the background wall velocity $\xi_w$ for different values of the phase transition strength $\alpha_N$ in the SM with low cut-off. The two different panels represents different friction parameterizations: on the left the results obtained considering the $\eta$ parameter constant, on the right the ones obtained with $\eta = \tilde{\eta} \, \phi^2 / T$.
• Figure 5: Same as \ref{['fig: SM_child_vs_parent_vw']} for the bag model, for $\delta a/a = 5.9 \times 10^{-3}$.