  We examine the spectrum of inflaton fluctuations resulting from any given
long period of exponential inflation. Infrared and ultraviolet divergences in
the inflaton dispersion summed over all modes do not appear in our approach. We
show how the scale-invariance of the perturbation spectrum arises. We also
examine the spectrum of scalar perturbations of the metric that are created by
the inflaton fluctuations that have left the Hubble sphere during inflation and
the spectrum of density perturbations that they produce at reentry after
inflation has ended. When the inflaton dispersion spectrum is renormalized
during the expansion, we show (for the case of the quadratic inflaton
potential) that the density perturbation spectrum approaches a mass-independent
limit as the inflaton mass approaches zero, and remains near that limiting
value for masses less than about 1/4 of the inflationary Hubble constant. We
show that this limiting behavior does not occur if one only makes the Minkowski
space subtraction, without the further adiabatic subtractions that involve time
derivatives of the expansion scale factor $a(t)$. We also find a parametrized
expression for the energy density produced by the change in $a(t)$ as inflation
ends. If the end of inflation were sufficiently abrupt, then the temperature
corresponding to this energy density could be very significant. We also show
that fluctuations of the inflaton field that are present before inflation
starts are not dissipated during inflation, and could have a significant
observational effect today. The mechanism for this is caused by the initial
fluctuations through stimulated emission from the vacuum.
