Cosmological phase transitions: from particle physics to gravitational waves, semi-analytically

TL;DR

This work develops a semi-analytic pipeline to predict gravitational-wave signals from cosmological first-order phase transitions by casting the finite-temperature effective potential into a quartic polynomial, including Daisy resummations and RG running. Using a classically scale-invariant $U(1)'$ model as a concrete testbed, the authors show how to parametrize the potential with high-temperature coefficients, project Daisy contributions onto polynomial bases (Legendre or Gram-Schmidt), and compute the tunneling action $S_3$ without solving bounce equations numerically at each point. They introduce an analytic method to determine the percolation temperature $T_p$, crucial for GW production, and quantify the impact of different approximations on $S_3$ and $T_p$, finding typical errors at the few-percent level for $S_3$ and sub-10-percent for $T_p$. The framework enables fast, large-scale parameter scans that remain faithful to full numerical results, offering a practical path to confront PTA-derived GW signals with particle-physics models and to guide synergies between gravitational-wave and laboratory probes.

Abstract

Motivated by the recent evidence of a stochastic gravitational wave background found by pulsar timing array experiments, we focus on one of the prime cosmological explanations, i.e. a supercooled first order phase transition. If confirmed, it would offer a unique opportunity to probe early Universe dynamics and the related physics beyond the Standard Model of particles and interactions. However, the prediction of the gravitational wave spectrum from a given particle physics scenario requires theoretically and computationally demanding methods. While several tools have been put forward to reduce uncertainties and automatize these computations, we study here the possibility to perform the full pipeline of computations semi-analytically in the $4D$ theory, thus avoiding computationally intensive simulations. Our approach yields accurate results that can be used in phenomenological studies and allow for an efficient exploration of the connection between the particle physics models and their cosmological predictions.

Figures (6)

• Figure 1: Effective potential without Daisy contributions using the full thermal function $J_b$ (red line) or its HT expansion from Eq. (\ref{['eq:HT_thermal_func']}) (dashed yellow line), for two different values of the gauge coupling $g'_0$ at the reference scale $\mu_0=g'_0\varphi_0$. The lower part of the plots shows the relative error, which starts to be large for large values of $\varphi/T$ where the HT expansion is not justified. The vertical asymptote corresponds to the point where the potential crosses zero.
• Figure 2: Comparison between the full Daisy contribution to the effective potential for the $U(1)^{\prime}$ model, shown as a solid red line, and the polynomial approximations. The lower panels show the relative error for each approximation, for $b=3$ (left) and $b=5$ (right). The dashed green line corresponds to the Gram-Schmidt procedure, while the dash-dotted blue line to the approximation in terms of Legendre polynomials.
• Figure 3: Bounce action for the $U(1)^{\prime}$ scale-invariant scenario using different approximations for the effective potential and the computation of $S_3(T)$. In both panels, the red solid line corresponds to the result using the full effective potential and computing the bounce action numerically with CosmoTransitions. The green dashed line corresponds to the effective potential in Eq. (\ref{['eq:Veff_GS_final']}), the dash-dotted blue one to $V_{\mathrm{eff}}$ in Eq. (\ref{['eq:Veff_Leg_final']}), and the dotted yellow line neglects the Daisy contribution. In the left panel we compare the impact of different approximations to $V_{\mathrm{eff}}$ on $S_3$, computed using the shooting method. In the right panel, we compare the values of $S_3$, obtained with the fit from Ref. Levi:2022bzt, for the polynomial potentials against the full result.
• Figure 4: Comparison of the result for $f_i(T,T')$ assuming radiation domination (blue), vacuum domination (red) or using the full Hubble expansion rate (black dash-dotted). The vertical dashed gray line represents the temperature below which vacuum energy dominates the expansion of the Universe. In the left panel radiation dominates the energy density of the Universe, while in the right one we have an intermediate case.
• Figure 5: Comparison between different approximations for the computation of the percolation temperature. The red line corresponds to the full numerical computation in the plots showing $T_p$ vs $g'_0$. The different colored lines correspond to the same approximations for the effective potential as in Fig. \ref{['fig:comparison_actions']}. In the upper panels we compute the action with a shooting method, while in the lower ones we use the action for the polynomial potentials from Ref. Levi:2022bzt. In the left panels the percolation temperature is found by integrating Eq. (\ref{['eq:percolation_full']}) numerically. Finally, in the right panels we instead solve Eq. (\ref{['eq:approx_percolation_final']}) to find $T_p$. The solid red line in the plot for the relative error compares the full solution found integrating against using Eq. (\ref{['eq:approx_percolation_final']}).