The hybrid inflation waterfall and the primordial curvature perturbation

TL;DR

This paper estimates the curvature perturbation generated during the hybrid inflation waterfall, analyzing its spectrum, and discusses implications for black hole abundance and inflation models, introducing a new criterion for transition sharpness.

Contribution

It provides a novel estimate of the waterfall contribution to curvature perturbation without assuming a specific inflaton potential, and introduces a criterion for the sharpness of the transition.

Findings

The spectrum peaks at a specific wavenumber and is negligible on cosmological scales.

Black hole bounds are satisfied if the curvaton mass exceeds the Hubble parameter.

A new 'end-of-inflation' formula and criterion for transition sharpness are proposed.

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 $\zeta=- (g^2 - \vev{g^2})$ 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 $\zeta$ from the rest of the waterfall, we are led to consider the use of the `end-of-inflation' formula, giving the contribution to $\zeta$ 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