Bubble velocities in local equilibrium from a pseudopotential

TL;DR

The paper addresses the challenge of predicting terminal bubble velocities during cosmological first-order phase transitions in local thermal equilibrium. It introduces a pseudopotential $\hat{V}(φ)$ derived from the finite-temperature potential $V(φ,T)$ and hydrodynamic temperature profiles, so that stationary configurations satisfy $\Delta \hat{V}=0$, avoiding direct solution of the scalar EOM or reliance on a fixed equation of state. Applied to a singlet-extended Standard Model, the method demonstrates sub-percent agreement with exact EOM results and reveals that static deflagrations are stable while detonations are unstable, with no stationary hybrids found in the explored space. This approach provides a robust, efficient tool for predicting bubble dynamics relevant to electroweak baryogenesis and gravitational-wave signals, and it can be extended to incorporate gradient effects beyond LTE if needed.

Abstract

We present a new method to estimate terminal bubble velocities during first-order phase transitions in a plasma in local equilibrium. The method relies on calculating the extrema of a modified potential function for the scalar field undergoing the transition. The shape of this function, which we refer to as the ``pseudopotential'', changes with the wall velocity, and if the dependence of the fluid temperature on scalar gradients is weak -- which is confirmed to hold with high accuracy in concrete examples -- the difference in pseudopotential between two appropriate extrema gives the net outward pressure acting on the bubble wall. It then follows that the correct terminal bubble velocities are those that lead to degenerate minima in the pseudopotential. This allows to compute bubble velocities without having to solve the equation of motion of the scalar field, and in contrast to other methods this can be done without relying on simplified equations of state for the plasma or without choosing a specific ansatz for the scalar field profile. We illustrate the method in a singlet extension of the Standard Model, computing the net outward pressure as a function of the wall velocity. We confirm the dip in outward pressure found in the literature for hybrid bubbles, which implies that stationary deflagrations are stable, while their detonation counterparts are unstable.

Figures (8)

• Figure 1: Different types of expanding bubbles with corresponding qualitative fluid velocity profile. The sphere represents the bubble wall moving in the direction of the arrows, and the blue contour depicts regions with nonzero fluid velocity. For deflagrations one has $v_w < c_s$, while hybrids and detonations satisfy $v_w > c_s$.
• Figure 2: Illustration of the physical meaning of the pseudopotential. Its extrema coincide with the minima $\phi_{\pm}$ of the ordinary potential on which the field is expected to settle ahead of/behind the bubble wall. The difference of the values of $\hat{V}(\phi)$ at the extrema gives the net outward pressure on the bubble wall. Changing the wall velocity $v_w$ changes the pseudopotential $\hat{V}(\phi)$ and with it the outward pressure, which is zero for stationary bubbles.
• Figure 3: Pseudopotential in the deflagration (left) and detonation (right) regime for $N=4$ and $\lambda_{HS} = 0.85$. The dark blue line indicates the steady state solution with $\Delta \hat{V} = 0$. For deflagrations, from top to bottom $T_+$ is $117.93$ GeV, $117.68$ GeV, $117.45$ GeV and $v_+$ is $-0.470$, $-0.444$, $-0.400$. For detonations $T_+ = T_{\text{nuc}} = 116.993$ GeV and $v_+$ is $-0.680$, $-0.735$, $-0.930$.
• Figure 4: Validity of taking $h^\prime(z) = 0$ in $T(h,h^\prime)$, demonstrated by explicit numerical calculation for $N=4$. The left figure shows the relative deviation between the physical wall velocity obtained from the pseudopotential and by solving Eq. \ref{['eq:differential_equation']} demanding $h^{\prime \prime}(z_\text{min})= 0$. The small deviation confirms that the temperature dependence on $h^\prime$ has a negligible effect. This is illustrated in the right plot, showing the dimensionless quantity $\Delta T/T(h,h^\prime)$ along the full profile distance, specifically for $\lambda_{HS} = 0.85$. The results indicate that the solution obtained from the pseudopotential provides an excellent approximation to the exact solution.
• Figure 5: Upper plot: Effective pressure as a function of the wall velocity for three different values of the coupling constant $\lambda_{HS}$. Solid lines indicate solutions where the extrema of $\hat{V}(h)$ are separated by a barrier, while dashed lines correspond to cases without a barrier, i.e. when one of the extrema is a maximum of $\hat{V}(h)$. The grey contour area depicts the segment in which hybrid solutions are expected to exist, the light grey hatched region indicates a region in which no solutions to the hydrodynamic matching constraints could be found. The physical solutions with $\Delta \hat{V}=0$ for deflagrations are stable, whereas the detonation solutions are expected to decay either to the left or the right. Stationary hybrid solutions cannot be found. Lower plot: Pseudopotential $\hat{V}(h)$ for three values of the bubble wall velocity and three choices of the coupling constant $\lambda_{HS}$.