GEFF: The Gradient Expansion Formalism Factory - A tool for inflationary gauge-field production

TL;DR

Inflationary gauge-field production can backreact on the background via nonlinear dynamics, and lattice simulations, while accurate, are computationally expensive. The paper presents GEFF, a Python package that implements the gradient expansion formalism (GEF) to evolve an infinite tower of ODEs for gauge-field bilinears with truncation at $n_{tr}$ and self-consistent checks against mode functions $A_\lambda(t,k)$ computed in tandem, regulated up to $k_{UV}(t)$. It ships with pre-defined models for pure axion inflation and fermionic axion inflation (FAI), and supports user-defined GEF models, along with tools to derive the tensor power spectrum and the induced gravitational-wave spectrum. Together these enable rapid exploration and benchmarking of inflationary gauge-field production scenarios while providing a pathway to lattice cross-checks.

Abstract

The GEFF - the Gradient Expansion Formalism Factory - is a new Python package designed to study gauge-field production during inflation. The package provides a framework to implement and use the gradient expansion formalism (GEF), a numerical technique devised to study the nonlinear dynamics associated with inflationary gauge-field generation. The GEF has already been applied in the context of axion inflation, and with the GEFF package, one can build on these results. The GEFF gives users access to ready-to-use model files for two scenarios of axion inflation: pure axion inflation, with the inflaton coupled to a pure Abelian gauge sector, and fermionic axion inflation, which assumes that the Standard Model (SM) hypercharge field is coupled to the inflaton, resulting in the production SM fermions via the Schwinger effect. The GEFF provides the user with methods to solve GEF equations, including an integrated error estimator and self-correction algorithm. Furthermore, users can implement their own GEF models, e.g., variations of axion inflation or related scenarios. The package also comes with tools to study the production of primordial gravitational waves induced by gauge fields. This is a starting guide for the GEFF, providing a high-level introduction to the GEF, installation instructions, and the basics for using this package.

Figures (4)

• Figure 1: A sketch of the algorithm behind the run method
• Figure 2: Results from run using pai with $\beta=15$ and $m=6.16\times 10^{-6} M_\mathrm{P}$. Panel a): The evolution of the energy densities extracted from sol. Panel b): The gauge-mode spectra $|A_\lambda(t,k)|$ extracted from spec shown for $\Delta N = -9$, $-3$, and $3$. In both plots, we use $\Delta N$ as the number of $e$-folds starting from the end of inflation expected from slow-roll dynamics.
• Figure 3: Results from run using fai_kh with $m=2\times 10^{-5} M_\mathrm{P}$. Panel a): Energy-density evolution for $\beta=50$. Panel b): Energy-density evolution for $\beta=60$. $\Delta N$ is defined as in Fig. \ref{['fig: evolution']}.
• Figure 4: Gravitational-wave spectra for the results of Figs \ref{['fig: evolution']} and \ref{['fig: fai-evol']}. We also show the spectrum for pai with $\beta=28$ and $m=6.16\times 10^{-6} M_\mathrm{P}$.