A density spike on astrophysical scales from an N-field waterfall transition

Abstract

Hybrid inflation models are especially interesting as they lead to a spike in the density power spectrum on small scales, compared to the CMB, while also satisfying current bounds on tensor modes. Here we study hybrid inflation with N waterfall fields sharing a global SO(N) symmetry. The inclusion of many waterfall fields has the obvious advantage of avoiding topologically stable defects for N>3. We find that it also has another advantage: it is easier to engineer models that can simultaneously (i) be compatible with constraints on the primordial spectral index, which tends to otherwise disfavor hybrid models, and (ii) produce a spike on astrophysically large length scales. The latter may have significant consequences, possibly seeding the formation of astrophysically large black holes. We calculate correlation functions of the time-delay, a measure of density perturbations, produced by the waterfall fields, as a convergent power series in both 1/N and the field's correlation function Δ(x). We show that for large N, the two-point function is 〈δt(x)δt(0)〉 ∝ Δ[superscript 2](|x|)/N and the three-point function is 〈δt(x)δt(y)δt(0)〉 ∝ Δ(|x−y|)Δ(|x|)Δ(|y|)/N[superscript 2]. In accordance with the central limit theorem, the density perturbations on the scale of the spike are Gaussian for large N and non-Gaussian for small N.