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.