ELENA: a software for fast and precise computation of first order phase transitions and gravitational waves production in particle physics models

TL;DR

ELENA delivers a fast, numerically stable implementation of the tunnelling potential formalism to compute first-order phase transitions in particle physics models and closes the loop by predicting the associated stochastic gravitational wave background. It provides a full end-to-end pipeline—from finite-temperature potentials and critical/milestone temperatures to tunnelling actions, vacuum fractions, thermal GW parameters, and the GW spectrum—without relying on the traditional bounce approach. The package introduces a vectorised, ODE-free method via espinosa.Vt_vec, calculates the percolation-based quantities, and extends the analysis beyond bag-model assumptions, enabling accurate model scans and robust GW predictions. In practice, ELENA integrates with CosmoTransitions models and PTArcade for model-building, parameter inference, and data-driven constraints, as demonstrated by MCMC studies against NANOGrav data and cross-validation with CosmoTransitions performance. These features collectively empower phenomenologists to systematically explore BSM scenarios with testable GW signatures across current and future experiments.

Abstract

We present ELENA (EvaLuator of tunnElliNg Actions), an open-source Python package designed to compute the full evolution of first-order phase transitions in the early Universe generated by particle physics models, taking into account several refinements that go beyond commonly assumed simplifications. The core of ELENA is based on a vectorized implementation of the tunnelling potential formalism, which allows for a fast computation of the finite-temperature tunnelling action. This, in turn, enables the sampling of the full range of temperatures where two phases coexist and the use of integral expressions that track the complete evolution of the transition, providing a comprehensive picture of it. In addition, ELENA provides all the tools to compute the resulting stochastic gravitational waves spectrum, allowing for the full chain of computations - from the Lagrangian parameter inputs to the final gravitational waves spectrum - in a fast and self-contained implementation.

Figures (25)

• Figure 1: Pictorial representation of the classical mechanical dual of the bounce equations of motion, \ref{['eq:bounce', 'eq:boundary']}. Upper panel: a classical particle at position $\phi$ is subject to the potential $-V(\phi)$, with a friction term that decreases with time $\rho$ (see \ref{['eq:bounce']}). The solution satisfying the boundary conditions in \ref{['eq:boundary']} amounts to finding the value of $\phi_0$ such that the particle, starting at rest from $\phi = \phi_0$ at $\rho = 0$, stops at $\phi = \phi_+$ for $\rho \rightarrow \infty$. Lower panel: the scalar field nucleates via an $O(d)$-symmetric field configuration, with value $\phi_0$ at $\rho = 0$ and $\phi_+$ at $\rho \rightarrow \infty$, where $\rho$ is the Euclidean radius in $d$ dimensions. The field profile at finite values of $\rho$ is determined by the solution $\phi_b(\rho)$ to \ref{['eq:bounce', 'eq:boundary']}.
• Figure 2: Example of the tunnelling potential formalism. Left panel: the tunnelling potential $V_t(\phi, \widetilde{\phi}_0)$ for different values of the $\widetilde{\phi}_0$ parameter, obtained by following the procedure outlined in Ref. Espinosa:2018hue; the blue line indicates the potential $V_t(\phi, \phi_0)$, obtained by choosing the value of $\widetilde{\phi}_0= \phi_0$ that minimises the action $S_{E,d} (\widetilde{\phi}_0)$. Gray lines represent configurations where the construction of the tunnelling potential breaks down, while orange lines are numerically viable configurations. For reference, the potential $V(\phi)$ is plotted as a black line. Right panel: the value of the action $S_{E,d}(\widetilde{\phi}_0)$ as a function of the $\widetilde{\phi}_0$ parameter, computed following \ref{['eq:action']}. $\phi_0^\textrm{min}$ is the smallest value of $\widetilde{\phi}_0$ for which the tunnelling can take place, that is, the smallest field value in the region to the right of the local barrier for which $V(\phi) < V(\phi_+)$. Values in gray correspond to unreliable $V_t$ constructions (cf. the gray lines in the left panel), while the blue line corresponds to numerically reliable computations. The inset box magnifies the curve around $\widetilde{\phi}_0 = \phi_0$, the value of $\widetilde{\phi}_0$ that minimises $S_{E,d}$.
• Figure 3: Shape of the finite temperature potential for the example points in \ref{['tab:input']} at different temperatures, corresponding to $T = 1.1 \cdot T_c$ (blue line), where $T_c$ is the critical temperature, $T_c$ (orange dashed-line), $T = (T_c - T_{min})/2$ (green line), where $T_{min}$ is the smallest temperature at which a barrier is present, and $T_{min}$ (red dot-dashed line). Additionally, when $T_{min}\neq 0$, we show the potential at $T=0$ as a purple line. Up to a global rescaling of $V$, these results are valid for an arbitrary energy scale (cf. \ref{['eq:scaling']}). The values of the temperatures shown in the legend correspond to the same ones chosen for $\phi$ and given in .
• Figure 4: Evolution of the relevant quantities related to tunnelling as a function of temperature for the "Fast" point in \ref{['tab:input']}. Upper-left: action $S_{E,3} / T$. Upper-right: tunnelling solution $\phi_0$. Centre-left: potential at true vacuum $V(\phi_-)$. Centre-right: potential at $\phi_0$, $V(\phi_0)$. Bottom-left: true vacuum $\phi_-$. Bottom-right: false vacuum $\phi_+$.
• Figure 5: Same as in \ref{['fig:action_fast']}, but for the "Slow" point in \ref{['tab:input']}.