  If the inflationary phase lasted longer than the minimal period, the length
scales observed today originate from modes that were smaller than the Planck
length during inflation. It was recently argued that this "trans-Planckian
problem" can never arise in a consistent string theory framework, which places
a stringent constraint on the energy scale of inflation, $V^{1/4}\lesssim 10^9$
GeV. In this paper, we show that this requirement corresponds to a very small
Hubble scale during inflation, $H_{\rm inf}\lesssim 1$ GeV, and therefore has
serious consequences on scenarios where the dark matter density was generated
by amplification of quantum fluctuations during inflation. We also present a
class of inflationary models which both satisfy the above limit for the scale
of inflation and are in perfect agreement with observational data.
