Supercool subtleties of cosmological phase transitions

Abstract

We investigate rarely explored details of supercooled cosmological first-order phase transitions at the electroweak scale, which may lead to strong gravitational wave signals or explain the cosmic baryon asymmetry. The nucleation temperature is often used in phase transition analyses, and is defined through the nucleation condition: on average one bubble has nucleated per Hubble volume. We argue that the nucleation temperature is neither a fundamental nor essential quantity in phase transition analysis. We illustrate scenarios where a transition can complete without satisfying the nucleation condition, and conversely where the nucleation condition is satisfied but the transition does not complete. We also find that simple nucleation heuristics, which are defined to approximate the nucleation temperature, break down for strong supercooling. Thus, studies that rely on the nucleation temperature $\unicode{x2014}$ approximated or otherwise $\unicode{x2014}$ may misclassify the completion of a transition. Further, we find that the nucleation temperature decouples from the progress of the transition for strong supercooling. We advocate use of the percolation temperature as a reference temperature for gravitational wave production, because the percolation temperature is directly connected to transition progress and the collision of bubbles. Finally, we provide model-independent bounds on the bubble wall velocity that allow one to predict whether a transition completes based only on knowledge of the bounce action curve. We apply our methods to find empirical bounds on the bubble wall velocity for which the physical volume of the false vacuum decreases during the transition. We verify the accuracy of our predictions using benchmarks from a high temperature expansion of the Standard Model and from the real scalar singlet model.

Figures (9)

• Figure 1: The energy density as a function of temperature for the benchmark point M2-BP1 defined in \ref{['tab:benchmarks-M2']}. The vertical dashed black line corresponds to $T_{\text{eq}}$ (defined in \ref{['eq:Teq']}). The energy density is dominated by $\rho_V$ at low temperatures and by $\rho_R$ at high temperatures.
• Figure 2: A 2D slice (with $z = 0.422$) of a simple 3D simulation of bubble nucleation in a unit volume. At the moment of plotting, the false vacuum fraction was $P_f \approx 0.7$ suggesting the onset of percolation. Indeed, a path through connected bubbles spanning from $y = 0$ to $y = 1$ is evident (see the green dashed curve), with the small gaps filled at slightly higher $z$ values. In general, the path spanning across the simulation volume may not be clear from a single 2D slice of the volume at the onset of percolation. The colour of a bubble corresponds to its overall radius, which is not equal to the radius of the bubble's cross-section in the $z=0.422$ plane depicted here.
• Figure 3: (a) The rate of change in physical volume of the false vacuum $\mathcal{V}_{\text{phys}}$, and (b) the physical volume of the false vacuum relative to its value at the completion (or final) temperature $\mathcal{V}_{\text{phys}}(T_f)$, for benchmark M1-BP4 (defined in \ref{['tab:benchmarks-M1']}) with $v_w = 1$. The physical volume of the false vacuum is seen to decrease slightly in the small shaded temperature window, yet increases during the rest of the transition, and continues to increase after the transition is complete. We expect finite regions of the Universe to remain in the false vacuum for such a transition, even though \ref{['eq:decreasingPhysicalVolume']} is satisfied for a range of temperatures. The vertical dashed lines are, in order from left to right: the completion temperature $T_f$ (black), the temperature $T_e$ for which $\mathcal{V}_t^{\text{ext}}(T) = 1$ (blue), and the percolation temperature $T_p$ (green). The physical volume decreases in the red shaded temperature window and increases everywhere else.
• Figure 4: An example of an action curve. This example is from a benchmark, M1-BP5, that we will define later in \ref{['sec:M1']}. Of our benchmarks in the two models that we investigate in \ref{['sec:models']}, this benchmark has the largest deviation from a quadratic action around $T_{\Gamma}$ (the dashed red line). The maximum nucleation rate occurs at $T_{\Gamma} \approx 10.25$ GeV, while the minimum of the action occurs at $T_{S_{\text{min}}} \approx 10.12$ GeV. Not displayed are the vertical asymptotes at $T=0$ and $T=T_c$ (the dashed black line). This U-shaped action curve is typical of supercooled transitions (see e.g. Refs. Megevand:2016lprCai:2017tmhKobakhidze:2017mru).
• Figure 5: The function $Y(T, 0)$ for benchmark M2-BP1 with $v_w = 0.220$. The horizontal dashed line marks the threshold $c_{p}/N^{\frac{1}{3}}(0)$, where $N(0) \approx 1$. The vertical dotted lines correspond to $T_{\text{eq}}$ (green) and $T_{\Gamma}$ (red). This benchmark has $Y(T_{\Gamma}, 0) < c_{p}/N^{\frac{1}{3}}(0)$, and hence is predicted to not yield percolation. Indeed, for M2-BP1 we numerically find $v_w \approx 0.223$ is required for percolation.