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 redshift these
short wavelength modes to cosmological scales in such a way that the
scale-invariance is not lost. This requires non-trivial background dynamics
before the onset of standard radiation-dominated cosmology; we demonstrate that
one possible solution is inflation with a sufficiently large Hubble rate, for
this slow roll is not necessary. In addition, we also show that if the
slow-roll condition is added to inflation with a large Hubble rate, then for
any power law modified dispersion relation quantum vacuum fluctuations become
nearly scale-invariant when they exit the Hubble radius.
