InflationEasy: A C++ Lattice Code for Inflation

Abstract

InflationEasy is a lattice code specifically developed for cosmological inflation. It simulates the nonlinear dynamics of a scalar field on a three-dimensional lattice in an expanding FLRW universe using finite-difference spatial derivatives. Based in part on the well-known LATTICEEASY, it incorporates several features tailored specifically to inflationary applications, including a nonperturbative $δN$ method to compute the curvature perturbation at the end of inflation $ζ$ directly from the lattice. In addition to the scalar sector, the code can also simulate scalar-induced gravitational waves, accounting for contributions generated both during inflation and during the subsequent horizon re-entry of scalar perturbations, and enabling the computation of the resulting gravitational-wave background. \texttt{InflationEasy} enables fully nonlinear studies of regimes with large fluctuations or nonperturbative non-Gaussianities, which lie beyond the reach of standard perturbation theory. It is applicable to a broad range of inflationary models, including those relevant for primordial black hole formation, gravitational-wave backgrounds, and large-scale structure.

Figures (3)

• Figure 1: Example of typical outputs from the lattice simulation. Top: Evolution of the spatial average of the inflaton field and its velocity as a function of the number of $e$-folds $N$. Middle: Left panel shows the evolution of the different energy components in the lattice; right panel displays the power spectrum of the curvature perturbation $\zeta$, computed both via the linear relation and the $\delta N$ method. Bottom: Left panel shows the probability distribution function (PDF) of $\zeta$; right panel shows a 2D slice of the final $\zeta$ field in position space. These outputs correspond to the default example in the code—namely, the ultra-slow-roll (USR) potential producing a peak in the power spectrum—run on a lattice with $N^3_{\rm pts} = 128^3$ points. These plots are generated using the example notebook plot.ipynb, included in the public release under the notebooks/ directory.
• Figure 2: Scalar-induced gravitational-wave pipeline implemented inInflationEasy. Starting from sub-horizon inflaton fluctuations evolved non-perturbatively on the lattice during inflation (a), the simulation yields the super-horizon curvature perturbation $\zeta$ at the end of inflation (b). This curvature field is then mapped to the Newtonian potential $\Phi$ on super-horizon scales using the initial relation given in \ref{['eq:zeta_to_phi']} (c), which provides the initial conditions for the post-inflationary evolution. The subsequent real-space evolution describes the horizon re-entry of scalar perturbations, yielding the sub-horizon Newtonian potential (d). The resulting gravitational-wave signal contains two contributions: an inflationary component, generated during inflation (GW $1$), and a post-inflationary component sourced at horizon re-entry (GW $2$).
• Figure 3: Post-inflationary dimensionless tensor power spectrum $\Delta_h^2(k)$ (left) and gravitational-wave energy density spectrum $\Omega_{\rm GW}(k)$ (right) in the default ultra-slow-roll (USR) example, obtained from the same simulation run used for \ref{['fig']}. The color scale indicates the number of e-folds $N=\log a$ from the beginning of the post-inflation simulation. These plots are generated using the example notebook plot.ipynb included in the public release.