The hybrid inflation waterfall and the primordial curvature perturbation

TL;DR

This paper develops a general treatment of the curvature perturbation generated during the linear era of the hybrid inflation waterfall, without committing to a specific inflaton potential. It shows that the waterfall-induced spectrum P_ζ(k) peaks at a scale k_* and is typically negligible on cosmological scales, with the peak amplitude and relevance depending on the ratio m/H and the transition dynamics. The authors derive an end-of-inflation formula to estimate the nonlinear contribution ζ_nl and apply the framework to SUSY-based models, finding that the cosmological black-hole bound is usually satisfied for m ≫ H but can be violated when m ≲ H. They highlight the crucial roles of non-Gaussianity, the peak scale k_*(t_end), and transition duration in determining the viability of hybrid inflation scenarios, and they outline necessary future work to refine the UV treatment and fully quantify the nonlinear regime.

Abstract

Without demanding a specific form for the inflaton potential, we obtain an estimate of the contribution to the curvature perturbation generated during the linear era of the hybrid inflation waterfall. The spectrum of this contribution peaks at some wavenumber $k=k_*$, and goes like $k^3$ for $k\ll k_*$, making it typically negligible on cosmological scales. The scale $k_*$ can be outside the horizon at the end of inflation, in which case $ζ=- (g^2 - \vev{g^2})$ with $g$ gaussian. Taking this into account, the cosmological bound on the abundance of black holes is likely to be satisfied if the curvaton mass $m$ much bigger than the Hubble parameter $H$, but is likely to be violated if $m\lsim H$. Coming to the contribution to $ζ$ from the rest of the waterfall, we are led to consider the use of the `end-of-inflation' formula, giving the contribution to $ζ$ generated during a sufficiently sharp transition from nearly-exponential inflation to non-inflation, and we state for the first time the criterion for the transition to be sufficiently sharp. Our formulas are applied to supersymmetric GUT inflation and to supernatural/running-mass inflation