Bubble Wall Velocity from Holography

Abstract

Cosmological phase transitions proceed via the nucleation of bubbles that subsequently expand and collide. The resulting gravitational wave spectrum depends crucially on the bubble wall velocity. Microscopic calculations of this velocity are challenging even in weakly coupled theories. We use holography to compute the wall velocity from first principles in a strongly coupled, non-Abelian, four-dimensional gauge theory. The wall velocity is determined dynamically in terms of the nucleation temperature. We find an approximately linear relation between the velocity and the ratio $Δ\mathcal{P}/\mathcal{E}$, with $Δ\mathcal{P}$ the pressure difference between the inside and the outside of the bubble and $\mathcal{E}$ the energy density outside the bubble. Up to a rescaling, the wall profile is well approximated by that of an equilibrium, phase-separated configuration at the critical temperature. We verify that ideal hydrodynamics provides a good description of the system everywhere except near the wall.

Figures (9)

• Figure 1: Energy density as a function of temperature and (inset) speed of sound for $\phi_Q=10, \phi_M=0.85$. The grey vertical line on the right indicates the critical temperature at which the PT takes place. The grey vertical line on the left indicates that $A$ and $B$ have the same temperature. Stable states are shown in solid blue, metastable ones in dashed brown, and unstable ones in dotted-dashed red.
• Figure 2: Different initial energy profiles for the same nucleation temperature $T_A$. In this and in subsequent plots we only show positive values of $z$ because we only consider states invariant under $z \to -z$.
• Figure 3: Snapshots of the energy density profile at different times for the initial state with $T_B=T_A$ shown as a dashed blue curve in Fig. \ref{['fig:initial_state']}.
• Figure 4: Same wall profiles as in Fig. \ref{['fig:E_vs_z']}, each shifted in $z$ by a different amount, to show that the wall profile remains constant in time.
• Figure 5: Energy profile at as a function of $\xi = z/t$ for different values of $t$.