Non-Gaussianity constrains hybrid inflation

TL;DR

This paper investigates whether the waterfall transition in hybrid inflation can convert isocurvature fluctuations into a nearly scale-invariant but highly non-Gaussian curvature perturbation, using the δN formalism to quantify the contribution $\zeta_w$ from the waterfall. The authors develop analytic approximations for tachyonic growth near the ridge, derive the statistics of $N$ and the resulting power spectrum and $f_{\mathrm{NL}}$, and translate these into constraints on the hybrid-model parameters, finding that non-Gaussianity can be tightly constrained unless the inflationary scale is very low. They also discuss the limits of the separate-universe picture in the presence of tachyonic preheating and emphasize the need for lattice simulations to determine the ultimate viability of the mechanism. The note added to the abstract indicates that nonlinear effects likely suppress the long-wavelength growth, casting doubt on the original claim of large $f_{\mathrm{NL}}$ and suggesting that the predicted signatures may not be realized in typical models.

Abstract

In hybrid inflationary models, inflation ends by a sudden instability associated with a steep ridge in the potential. Here we argue that this feature can generate a large contribution to the curvature perturbation on observable scales. This contribution is almost scale-invariant but highly non-Gaussian. The degree of non-Gaussianity can exceed current observational bounds, unless the inflationary scale is extremely low or the hybrid potential contains very large coupling constants. Non-linear effects on small scales may quench the non-Gaussian signal, and while we find no compelling evidence that this occurs, full lattice simulations are required to definitively address this issue. Note added: We now believe that nonlinear effects will invalidate the original computation in this paper essentially instantaneously after the short-wavelength modes reach the minimum of their potential. This means that the mechanism described in this paper will not lead to appreciatable curvature perturbations on long wavelengths, and no useful constraints on hybrid inflation will result. We have inserted a brief calculation on p2 of this manuscript to explain this fact, but have otherwise left the manuscript unchanged.

Figures (2)

• Figure 1: $\sigma_{\mathrm{w}}^2/(H/\mu)^2$ plotted as a function of the RMS initial fluctuation $\sigma_\chi$ in $\chi$. The green dots are computed numerically for a Gaussian ensemble of trajectories, using the potential e:numV and parameter values e:numVparams. The lines are semi-analytic estimates, obtained by numerically integrating (\ref{['eq:n-one']}-\ref{['eq:n-three']}). The solid red line uses a Gaussian distribution of initial conditions of variance $\sigma_\chi^2$, while the dashed blue line employs a uniform distribution of varaince $\sigma_\chi^2$. The final variance depends weakly on the form of the initial distribution, and varies extremely slowly over many decades of the initial variance.
• Figure 2: The skew of ${\zeta_{\mathrm{w}}}$ plotted as a function of the RMS initial fluctuation $\sigma_\chi$ in $\chi$. The green dots are computed numerically for a Gaussian ensemble of trajectories using the potential e:numV and parameter values e:numVparams. The lines are semi-analytic estimates, obtained by numerically integrating (\ref{['eq:n-one']}-\ref{['eq:n-three']}). The solid red line uses a Gaussian distribution of initial conditions of variance $\sigma_\chi^2$, while the dashed blue line employs a uniform distribution of variance $\sigma_\chi^2$. The final skew depends weakly on the form of the initial distribution, and varies very little over many decades of variation in the initial variance.