Density Perturbations in Hybrid Inflation Using a Free Field Theory Time-Delay Approach

Abstract

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.