Vacuum fluctuations in theories with deformed dispersion relations
  We examine vacuum fluctuations in theories with modified dispersion relations
which represent dimensional reduction at high energies. By changing units of
energy and momentum we can obtain a description rendering the dispersion
relations undeformed and transferring all the non-trivial effects to the
integration measure in momentum space. Using this description we propose a
general quantization procedure, which should be applicable whether or not the
theory explicitly introduces a preferred frame. Based on this scheme we
evaluate the power spectrum of quantum vacuum fluctuations. We find that in
{\it all} theories which run to 2 dimensions in the ultraviolet the vacuum
fluctuations, in the ultraviolet regime, are scale-invariant. This is true in
flat space but also for "inside the horizon" modes in an expanding universe. We
spell out the conditions upon the gravity theory for this scale-invariance to
be preserved as the modes are frozen-in outside the horizon. We also digress on
the meaning of dimensionality (in momentum and position space) and suggest that
the spectral index could itself provide an operational definition of
dimensionality.
