Density Perturbations in Hybrid Inflation Using a Free Field Theory
  Time-Delay Approach
  We introduce a new method for calculating density perturbations in hybrid
inflation which avoids treating the fluctuations of the "waterfall" field as if
they were small perturbations about a classical trajectory. We quantize only
the waterfall field, treating it as a free quantum field with a time-dependent
$m^2$, which evolves from positive values to tachyonic values. Although this
potential has no minimum, we think it captures the important dynamics that
occurs as $m^2$ goes through zero, at which time a large spike in the density
perturbations is generated. We assume that the time-delay formalism provides an
accurate approximation to the density perturbations, and proceed to calculate
the power spectrum of the time delay fluctuations. While the evolution of the
field is linear, the time delay is a nonlinear function to which all modes
contribute. Using the Gaussian probability distribution of the mode amplitudes,
we express the time-delay power spectrum as an integral which can be carried
out numerically. We use this method to calculate numerically the spectrum of
density perturbations created in hybrid inflation models for a wide range of
parameters. A characteristic of the spectrum is the appearance of a spike at
small length scales, which can be used to relate the model parameters to
observational data. It is conceivable that this spike could seed the formation
of black holes that can evolve to become the supermassive black holes found at
the centers of galaxies.
