The Trans-Planckian Problem for Inflationary Cosmology Revisited
  We consider an inflationary universe model in which the phase of accelerated
expansion was preceded by a non-singular bounce and a period of contraction
which involves a phase of deceleration. We follow fluctuations which exit the
Hubble radius in the radiation-dominated contracting phase as quantum vacuum
fluctuations, re-enter the Hubble radius in the deflationary period and
re-cross during the phase of inflationary expansion. Evolving the fluctuations
using the general relativistic linear perturbation equations, we find that they
exit the Hubble radius during inflation not with a scale-invariant spectrum,
but with a highly red spectrum with index $n_s = -3$. We also show that the
back-reaction of fluctuations limits the time interval of deflation. Our toy
model demonstrates the importance for inflationary cosmology both of the
trans-Planckian problem for cosmological perturbations and of back-reaction
effects . Firstly, without understanding both Planck-scale physics and the
phase which preceded inflation, it is a non-trivial assumption to take the
perturbations to be in their local vacuum state when they exit the Hubble
radius at late times. Secondly, the back-reaction effects of fluctuations can
influence the background in an important way.
