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 field's correlation
function $\Delta(x)$. We show that for large $N$, the two-point function is
$<\delta t({\bf x})\,\delta t({\bf 0})>\,\propto\Delta^2(|{\bf x}|)/N$ and the
three-point function is $<\delta t({\bf x})\,\delta t({\bf y})\,\delta t({\bf
0})>\,\propto\Delta(|{\bf x}-{\bf y}|)\Delta(|{\bf x}|)\Delta(|{\bf y}|)/N^2$.
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 $N$.
