Ultra-Slow-Roll Inflation on the Lattice II: Nonperturbative Curvature Perturbation

TL;DR

This work advances nonperturbative predictions of early-universe curvature perturbations by extracting fully nonlinear ζ from lattice simulations of ultra-slow-roll inflation using the δN formalism. It reveals that nonlinear φ→ζ mapping significantly enhances the positive tail of the ζ distribution, while the power spectrum remains modestly affected except in extreme cases. By contrasting with perturbative gauge-change methods, it quantifies where perturbation theory breaks down and demonstrates that an effective constant-η background can capture the bulk ζ statistics. The results are crucial for robust predictions of primordial black holes and scalar-induced gravitational waves from inflation, and establish a lattice-based framework for future multi-field and tensorial extensions.

Abstract

Building on the recent lattice simulations of ultra-slow-roll (USR) dynamics presented in arXiv:2410.23942, we investigate the role of the nonlinear relation between the inflaton field configuration and the curvature perturbation $ζ$, the key observable after inflation. Using a nonperturbative $δN$ approach applied to the lattice output, we generate fully nonlinear three-dimensional maps of $ζ$. This calculation captures both the non-Gaussianity arising from the nonlinear mapping between $φ$ and $ζ$, and the intrinsic non-Gaussianity generated around Hubble crossing by the nonlinear field dynamics, which is neglected in stochastic approaches. We find that the nonlinear mapping has a profound impact on the statistics, significantly enhancing the positive tail of the $ζ$ probability distribution, with important implications for observable quantities. A central part of this work is the comparison with the standard perturbative treatment based on a gauge transformation, which allows us to quantify when and how the perturbative picture breaks down as fluctuations grow large. Together with arXiv:2410.23942, this work sets the basis for robust, nonperturbative predictions of primordial black hole production and scalar-induced gravitational wave emission from inflation using lattice simulations.

Figures (10)

• Figure 1: Graphical representation of the change of gauge relationship \ref{['eq:change-gauge']} for the fiducial background \ref{['eq:fiducial-background']} for $\eta=-0.5$. The change of gauge is performed at $N_*=2$ and the three plots correspond to different values of the scalar field in the flat gauge. See the main text for more details.
• Figure 2: Power spectrum of the comoving curvature perturbation $\zeta$ from the lattice simulation. Full lines represent the nonperturbative lattice results obtained with the $\delta N$ approach, dashed lines the linearized perturbation $\zeta_{\rm lin}=-H\delta\phi/\dot\phi$ and the dots show results from the logarithmic relation in Eq. \ref{['eq:log']} applied to $\zeta_{\rm lin}$. Different colors correspond to different lattice simulations presented in Paper Caravano:2024moy, ranging from ${\cal P}_{\zeta,\rm tree}^{\rm max} =10^{-4}$ to ${\cal P}^{\rm max}_{\zeta,\rm tree} =1$, where $\cal P_{\zeta,\rm tree}$ is the tree-level power spectrum.
• Figure 3: 1-point probability density function (PDF) of the comoving curvature perturbation $\zeta$. Different columns correspond to different $\mathcal{P}^{\rm max}_{\zeta,\rm tree}$ ranging from $10^{-3}$ to $1$, while the rows refer to the three different cases. Full lines represent the nonperturbative lattice results obtained with the $\delta N$ approach, black triangles correspond to the linearized perturbation $\zeta_{\rm lin}=-H\delta\phi/\dot\phi$, and the dots show results from the logarithmic relation in Eq. \ref{['eq:log']} applied to $\zeta_{\rm lin}$. Vertical gray lines indicate the value $\zeta=2$, where Eq. \ref{['eq:log']} diverges if applied to $\zeta_{\rm lin}$.
• Figure 4: Snapshots from the lattice simulations. For each panel, the left sub-panel shows the curvature perturbation obtained from the linear change of gauge $\zeta_{\textrm{lin}}=-H {\delta\phi}/{\dot\phi}$, while the right sub-panel shows the nonperturbative curvature perturbation $\zeta$. Different lines show the three different cases, while the columns are for different values of the tree-level power spectrum. The green regions indicate the trapped patches. Given the wide range of values of $\zeta$, to ease visualization, we only show positive values of $\zeta$ and set negative values to zero in this figure. In Fig. \ref{['fig:snapshots_negative']}, we show the negative values for completeness. In Fig. \ref{['fig:3d_snapshots']}, we show a 3D visualization of these snapshots for $\mathcal{P}_{\zeta,\rm tree}^{\rm max} = 1$ (right column of this figure).
• Figure 5: Same as Fig. \ref{['fig:snapshots']}, but for negative values of $\zeta$. In Fig. \ref{['fig:3d_snapshots']}, we show a 3D visualization of these snapshots for the case $\mathcal{P}_{\zeta,\rm tree}^{\rm max} = 1$ (right column of this figure).