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

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 NN waterfall fields sharing a global SO(N)SO(N) symmetry. The inclusion of many waterfall fields has the obvious advantage of avoiding topologically stable defects for N>3N>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/N1/N and the field's correlation function Δ(x)Δ(x). We show that for large NN, the two-point function is 〈δt(x)δt(0)〉∝Δ2(|x|)/N and the three-point function is 〈δt(x)δt(y)δt(0)〉∝Δ(|x−y|)Δ(|x|)Δ(|y|)/N2. In accordance with the central limit theorem, the density perturbations on the scale of the spike are Gaussian for large NN and non-Gaussian for small NN.