Semi-Analytic Calculation of the Gravitational Wave Signal From the Electroweak Phase Transition for General Quartic Scalar Effective Potentials

TL;DR

This work develops a semi-analytic framework to predict gravitational waves from a strongly first-order electroweak phase transition driven by a SM-like quartic scalar potential. By leveraging an analytic approximation to the three-dimensional Euclidean action, the authors derive closed-form expressions for the tunneling temperature $T_t$ and the GW parameters $\alpha$ and $\beta/H_t$ as functions of the potential parameters, including a temperature-independent cubic term $e$ that can substantially boost GW power. They map the parameter space, apply the formalism to SM+singlet and SM+triplet models, and show that singlet-induced cubic couplings can enhance detectability by LISA or BBO while preserving SM-like phenomenology. The results provide a versatile, rapid tool for connecting high-energy model-building with prospective GW observations, aiding discrimination of electroweak-scale new physics.

Abstract

Upcoming gravitational wave (GW) detectors might detect a stochastic background of GWs potentially arising from many possible sources, including bubble collisions from a strongly first-order electroweak phase transition. We investigate whether it is possible to connect, via a semi-analytical approximation to the tunneling rate of scalar fields with quartic potentials, the GW signal through detonations with the parameters entering the potential that drives the electroweak phase transition. To this end, we consider a finite temperature effective potential similar in form to the Higgs potential in the Standard Model (SM). In the context of a semi-analytic approximation to the three dimensional Euclidean action, we derive a general approximate form for the tunneling temperature and the relevant GW parameters. We explore the GW signal across the parameter space describing the potential which drives the phase transition. We comment on the potential detectability of a GW signal with future experiments, and physical relevance of the associated potential parameters in the context of theories which have effective potentials similar in form to that of the SM. In particular we consider singlet, triplet, higher dimensional operators, and top-flavor extensions to the Higgs sector of the SM. We find that the addition of a temperature independent cubic term in the potential, arising from a gauge singlet for instance, can greatly enhance the GW power. The other parameters have milder, but potentially noticeable, effects.

Figures (4)

• Figure 1: In each plot the labels for the x-axis denotes the varied parameter in blue and red (dashed), respectively. The green line is the (constant) SM value. The ranges for the varied parameters were chosen arbitrarily, to show the overall behavior. $e$ is in GeV, while the other parameters are dimensionless. Note the pole as $E$ is varied, as described. On the left-hand side, the blue line (varying $e$) closely follows the green one (SM value), for the range shown.
• Figure 2: A plot of the $\alpha-\beta/H$ plane with $\lambda$ and $D$ varying. The red line is the SM with $\lambda$ in the allowed range. The darker blue region is an order of magnitude greater and less than the SM value for $D$. The lighter blue region is a further order of magnitude greater and less. BBO and LISA rough sensitivity regions are shown as the lighter and darker red shading, respectively.
• Figure 3: A plot of the $\alpha-\beta/H$ plane with $\lambda$ and $E$ varying. The red line is the SM with $\lambda$ in the allowed range. The darker blue region is an order of magnitude greater and less than the SM value for $E$. The lighter blue region is a further order of magnitude greater and less. BBO and LISA rough sensitivity regions are shown as the lighter and darker red shading, respectively.
• Figure 4: A plot of the $\alpha-\beta/H$ plane with $\lambda$ and $e$ varying. The dark blue region is for $e > 0$, while the light blue shading indicates the first solution for $\lambda$ when $e < 0$, and the green the second solution. BBO and LISA rough sensitivity regions are shown as the lighter and darker red shading, respectively. The tail of the green region, extending to very large $\alpha$, may have errors in the calculation of $\beta/H_t$ due to the approximate formula. However, the points remain in the LISA region.