Intrinsic non-Gaussianity of ultra slow-roll inflation

TL;DR

The paper tackles the non-Gaussian tail of the curvature perturbation $\zeta$ in inflation with a transient USR phase that creates a localized peak in $\mathcal{P}_\zeta(k)$. It develops a three-step framework: (i) compute the tree-level power spectrum and bispectrum of the inflaton perturbation $\delta\phi$ in the $\delta\phi$-gauge via the in-in formalism, (ii) construct a non-Gaussian lattice realization that matches these statistics, and (iii) map to $\zeta$ using the $\delta N$ formalism to obtain the non-Gaussian PDF of $\zeta$. The analysis shows that intrinsic non-Gaussianities of $\delta\phi$, encoded in the intrinsic bispectrum $\mathcal I$, can be comparable to or larger than the non-Gaussianity arising from nonlinear $\delta\phi$ to $\zeta$ mapping (term $\mathcal N$), with their relative importance depending on the sharpness and duration of the USR→CR transition. The results emphasize that perturbative approaches may fail to capture the full tail in certain regimes and motivate non-perturbative methods for reliable PBH abundance estimates, while providing a framework that can incorporate higher-order non-Gaussianities and generalize to broader USR-like dynamics.

Abstract

We study the non-Gaussian tail of the curvature fluctuation, $ζ$, in an inflationary scenario with a transient ultra slow-roll phase that generates a localized large enhancement of the spectrum of $ζ$. To do so, we implement a numerical procedure that provides the probability distribution of $ζ$ order by order in perturbation theory. The non-Gaussianities of $ζ$ can be shown to arise from its non-linear relation to the inflaton fluctuations and from the intrinsic non-Gaussianities of the latter, which stem from its self interactions. We find that intrinsic non-Gaussianities, which have often been ignored to estimate the abundance of primordial black holes in this kind of scenario, are important. The relevance of the intrinsic contribution depends on the rapidity with which the transient ultra slow-roll phase occurs, as well as on its duration. Our method cannot be used accurately when the perturbative in-in formalism fails to apply, highlighting the relevance of developing fully non-perturbative approaches to the problem.

Figures (7)

• Figure 1: Consistency relation for the sharp (left), intermediate (center) and smooth (right) cases in Table \ref{['table:models']}, obtained by comparing the left-hand side of Eq. (\ref{['eq:consistency']}) (red crosses) and the right-hand side involving the derivative of the power spectrum (solid black line). The agreement implies that we have considered all interactions that are relevant (at least) in the squeezed limit.
• Figure 2: Left panel: Potentials obtained by varying $\delta$ between $0.1$ (green) and $0.9$ (blue), for $\Delta N=2$. Right panel: Potentials obtained by varying $\Delta N$ between $1.1$ (green) and $2$ (blue), for $\delta=0.5$. We have set $\eta_{\rm USR}=4$ and $\eta_{\rm CR}=-1$ for both panels.
• Figure 3: Colored contours correspond to the peak value of $\log_{10}\mathcal{P}_\zeta$. Dashed, solid and dot-dashed black lines are the contours $\mathcal{I}/\mathcal{N}=-0.5,0,0.5$, respectively. Left panel shows the dependence of these two quantities on $\Delta \eta$ and $\delta$, fixing $\Delta N=1.7$. Right panel shows their dependence on $\Delta N$ and $\delta$, fixing $\Delta\eta = 1$. The three white spots correspond to the three cases described in Table \ref{['table:models']}, which indeed have a similar peak height in their power spectrum (Figure \ref{['fig:corrs_I']}, right panel). The assumption of Gaussian $\delta\phi$ is only valid sufficiently close to the solid black line.
• Figure 4: Left panel: Contributions to the three-point function of $\zeta$ arising from the intrinsic non-Gaussianity of $\delta\phi$ (blue) and its nonlinear relation to $\zeta$ (red) rescaled by the factor $k^6$ for the sharp (solid), intermediate (dashed) and smooth (dot-dashed) examples in Section \ref{['sec:num_model']}. The vertical lines denote the approximate range of momenta probed by the lattice simulation presented in Section \ref{['sec:lattice']}. Right panel: Power spectra for the same examples as in the left panel. We denote the location of the peak in the spectra by $k_p$.
• Figure 5: Left panel: Two-dimensional slice of one realization of the intermediate transition example in Section \ref{['sec:num_model']} for the Gaussian contribution to $\tilde{\varphi}$ (i.e. $\tilde{\varphi}^{(2)}$). Right panel: Same realization as in the left panel for the full variable $\tilde{\zeta}$. Due to the strong non-Gaussian tail, hot spots are pushed to larger values.