On the Maximal Strength of a First-Order Electroweak Phase Transition and its Gravitational Wave Signal

TL;DR

The paper investigates how strong a first-order electroweak phase transition can be while still completing in the early universe, accounting for significant supercooling and the potential domination of vacuum energy on cosmic expansion. It develops a comprehensive framework for nucleation, growth, percolation, and reheating that remains valid when radiation no longer dominates, and applies it to polynomial BSM potentials to bound Tp and predict the gravitational-wave signal. The main findings are that percolation restricts extreme supercooling, the GW signal is dominated by plasma sound waves and turbulence with a peak frequency f ≳ 10^{-4} Hz, and that long-lasting sound waves are typically not realized, reducing the predicted GW amplitude relative to prior estimates. The results are illustrated in two concrete models (a |H|^6/Λ^2 EFT and a real singlet extension) and briefly discussed for conformal/dilaton-like potentials, highlighting implications for LISA detectability and collider probes.

Abstract

What is the maximum possible strength of a first-order electroweak phase transition and the resulting gravitational wave (GW) signal? While naively one might expect that supercooling could increase the strength of the transition to very high values, for strong supercooling the Universe is no longer radiation-dominated and the vacuum energy of the unstable minimum of the potential dominates the expansion, which can jeopardize the successful completion of the phase transition. After providing a general treatment for the nucleation, growth and percolation of broken phase bubbles during a first-order phase transition that encompasses the case of significant supercooling, we study the conditions for successful bubble percolation and completion of the electroweak phase transition in theories beyond the Standard Model featuring polynominal potentials. For such theories, these conditions set a lower bound on the temperature of the transition. Since the plasma cannot be significantly diluted, the resulting GW signal originates mostly from sound waves and turbulence in the plasma, rather than bubble collisions. We find the peak frequency of the GW signal from the phase transition to be generically $f \gtrsim 10^{-4}$ Hz. We also study the condition for GW production by sound waves to be long-lasting (GW source active for approximately a Hubble time), showing it is generally not fulfilled in concrete scenarios. Because of this the sound wave GW signal could be weakened, with turbulence setting in earlier, resulting in a smaller overall GW signal as compared to current literature predictions.

Figures (19)

• Figure 1: The normalized size $H_{\mathrm{V}} \,a(T')\, r(T,\, T')$ of a bubble nucleated at $T' = 5\, T_V$, as a function of $T/T_V$, for pure radiation domination \ref{['eq:RV']} (red dot-dashed line), the approximate vacuum domination solution \ref{['eq:r_RVD']} (green dashed line), and the exact \ref{['eq:RV']} (solid blue line) evolution from radiation to vacuum domination. We also show the size of a bubble nucleated at $T = T_{V}$ and growing in vacuum \ref{['eq:r_VD']} (purple dashed line). We have verified these behaviours in all four cases using the specific models from Section \ref{['sec:models']}.
• Figure 2: Detonation plasma velocity profile $v(\xi)$ for various values of $\alpha$ and $v_w$.
• Figure 3: The energy distribution $\mathcal{E}_B(t_p,R)=dnR^3/\int dn R^3$ (blue solid), bubble size distribution $dnR/\int dn R$ (green dashed) and number density $dn/\int dn$ (red dotted) as functions of the bubble size $R$ normalised to $R_{\rm MAX}$ (the maximum of the energy distribution), all computed at the nucleation temperature. The thin vertical line indicates $R_*/R_{\rm MAX}$, where $R_*$is the mean bubble separation. The left panel shows the example with the strongest transition where percolation can still be possible, namely $\Lambda = 545 \ {\rm GeV}$ and $T_p < T_{V}$, while the right panel shows the strongest transition for which percolation is assured, corresponding to $\Lambda = 545.7 \ {\rm GeV}$ and $T_p > T_{V}$ for the EFT model from Section \ref{['subsec:H6']}.
• Figure 4: The nucleation temperature $T_n$ (orange solid line), the percolation temperature $T_p$ (purple solid line) and the temperature $T_V$ (green solid line) below which vacuum energy dominates the expansion of the Universe. The temperature $T_p^{\rm RD}$ (red dashed line) and $T_p^{\rm VD}$ (blue dash-dot line) respectively show the percolation temperature obtained neglecting the vacuum energy contribution to the expansion, and using the approximation \ref{['eq:Hubbleapprox']}. The dark grey area is excluded by our percolation criterion \ref{['eq:falsevacuumvol']}, while in the light gray area percolation is questionable, as the criterion is only satisfied below $T_p$. Vertical lines show the projected reach of various detection methods: The green lines indicate values of $\Lambda$ to which the HL-LHC will be sensitive at the $3$- (solid) and 2-$\sigma$ (dashed) level respectively, and the blue lines show the reach of LISA from sound waves (dashed line) and turbulence (solid line).
• Figure 5: Values of $\Gamma/H^4$ (green dashed line), the number $N$ of bubbles per horizon \ref{['eq:T_n']} (red dashed line) and $I(T)$ (see \ref{['eq:prob_false_vacuum']} and \ref{['eq:prob_false_vacuum_2']}), as a function of $T$ for $\Lambda = 545 \ {\rm GeV}$, corresponding to the strongest transition where percolation can still be possible (left panel) and $\Lambda = 545.7 \ {\rm GeV}$, corresponding to the strongest transition for which percolation is assured (right panel). The vertical lines show (from left to right) the temperatures $T_p$, $T_V$ and $T_n$.