A Density Spike on Astrophysical Scales from an N-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 $N$ 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 fields correlation function $Delta(x)$. We show that for large $N$, the two-point function is $<delta t({bf x}),delta t({bf 0})>,proptoDelta^2(|{bf x}|)/N$ and the three-point function is $<delta t({bf x}),delta t({bf y}),delta t({bf 0})>,proptoDelta(|{bf x}-{bf y}|)Delta(|{bf x}|)Delta(|{bf y}|)/N^2$. 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 $N$.