Modified dispersion relations, inflation, and scale invariance

For a certain type of modified dispersion relations, the vacuum quantum state for very short wavelength cosmological perturbations is scale-invariant and it has been suggested that this may be the source of the scale-invariance observed in the temperature anisotropies in the cosmic microwave background. We point out that for this scenario to be possible, it is necessary to red-shift these short wavelength modes to cosmological scales in such a way that the scale-invariance is not lost. This can be done by inflation with a sufficiently large Hubble rate, without any requirement for slow roll. We also show that in the case of slow-roll inflation, modes that start in their vacuum quantum state will become nearly scale-invariant when they exit the Hubble radius for any power law modified dispersion relation