Trans-Planckian censorship, inflation, and dark matter

If the inflationary phase lasted longer than the minimal period, the length scales observed today originated 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 less than or similar to 10(9) GeV. In this paper, we show that this requirement corresponds to a very small Hubble scale during inflation, H-inf less than or similar to 1 GeV, and therefore has serious consequences on scenarios in which 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.